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SPH3UI Gr.11 Physics Name:

motion graphs

motion equations

relative motion

projectiles

Lesson #1 – Introduction to Motion

Pre-Lesson Notes

Unit: Kinematics/Motion Date: Concept: Uniform Motion

Definitions:Uniform Motion

Motion(a.k.a. Kinematics)

Position

Distance

Displacement

Vector

Scalar

Equations:

Explanation:

SPH3U: Introduction to Motion

Welcome to the study of physics! As young physicists you will be making measurements and observations, looking for patterns, and developing theories that help us to describe how our universe works. The simplest measurements to make are position and time measurements which form the basis for the study of motion.

A: Uniform MotionYou will need a motorized physics car, a pull-back car, a measuring tape or a ruler, and a stopwatch.

1. Which object moves in the steadiest manner: the buggy or the pull-back car? How did you decide?

Recorder: __________________Manager: __________________Speaker: __________________

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0 1 2 3 4 5

2

2. Excitedly, you show the buggy to a friend and mention how its motion is very steady or uniform. Your friend, for some reason, is sceptical. Describe some simple measurements you could make which would convince your friend that the motion of the buggy is indeed very steady (don’t have to do them!)

3. We call the motion of the buggy uniform motion. Use your ideas from the previous question to help write a definition for uniform motion. (Danger! Do not use the words speed or velocity in your definition!) When you’re done, write this on your whiteboard – you will share this later.

B: Tracking Motion – Position and TimeThe purpose of this investigation is to track the location of a moving object. Use a battery operated physics buggy or a pull-back car, measuring tape or a ruler, and a stopwatch.

1. Discuss with your group a process that will allow you to track the position of the car at various times. Draw a simple picture and illustrate the quantities to be measured. Describe this process as the procedure for your experiment.

Physicists define the position of an object that lies on a line as the distance between a reference point on the object and some other reference point, or origin, which is also on that line. Usually the position of an object along a horizontal line is positive along one side of the origin of the origin and negative if it lies on the other – but this sign convention is really a matter of choice. Choose your sign convention such that your position measurements will all be positive.

2. Move bags and stools and conduct your experiment. It may be useful to measure a floor tile and count tiles if you don’t have a tape measure. Record your data below.

Position ( m)

Time (s)

3. Explain how you can tell just by looking at the data if the motion is uniform.

3

4. Plot your data on a graph with position on the vertical axis and time on the horizontal axis.

5. The points should form a nearly straight line. Draw the best straight line that you can to represent your data. (Never draw a zig-zag pattern for data that appears liner!) What does the straightness of the graph tell you about the motion?

6. Calculate the slope of the graph and explain what the slope is describing about the object’s motion (please include the units!). Reminder: slope = rise / run.

7. Explain how you could predict (without using a graph) where would the car be found 1.0 s after your last measurement.

Lesson #2 – Interpreting Position-Time (DT) Graphs

Pre-Lesson Notes

Unit: Kinematics/Motion Date: Concept: Position Time (DT) Graphs

4

What quantity does the slope of the line on the DT give?

Explanation:

Details / Tricks

SPH3U: Interpreting Position-Time Graphs

Today you will learn how to relate position-time graphs to the motion they represent. We will do this using a computerized motion sensor which measures one-dimensional horizontal motion. The origin is at the sensor (position = 0) and the direction away from the face of the sensor is set as the positive direction.

A: Interpreting Position Graphs1. (Work individually) For each description of a person’s motion listed below, sketch your prediction for what

you think the position-time graph would look like. Use a dashed line for your predictions. Note that in a sketch of a graph we don’t worry about exact values, just the correct general shape. Try not to look at your neighbours predictions, but if you’re not sure how to get started, ask a group member for some help.

(a) Standing still, close to the sensor

(b) Standing still, far from the sensor


0 1 2 3 4 5

(0,0) Time (s)

Posi

tion

(m)

(0,0) Time (s)

Posi

tion

(m)

5

(c) Walking slowly away from the sensor at a steady rate.

(d) Walking quickly away from the sensor at a steady rate.

(e) Walking slowly towards the sensor at a steady rate

(f) Walking quickly towards the sensor at a steady rate.

2. (Work together) Compare your predictions with your group members and discuss any differences. Don’t worry about making changes. On a white board, quickly sketch your six predictions large enough so the whole class can see.

3. (As a class) Your group’s speaker is the official “walker”. The computer will display its results for each situation. Record the computer results on the graphs above using a solid line. Note that we want to smooth out the bumps and jiggles in the computer data which are a result of lumpy clothing, swinging arms, and the natural way our speed changes during our walking stride.

4. (Work together) Explain the errors in the following predictions. For situation (a) a students predicts: For situation (d) the student

says: “Look how long the line is – she travels far in a small amount of time. That means she is going fast.”

5. Describe the difference between the two graphs made by walking away slowly and quickly.

6. Describe the difference between the two graphs made by walking towards and away from the sensor.

B: The Position Prediction ChallengeNow for a challenge! From the description of a set of motions, can you predict a more complicated graph?

A person starts 2.0 m in front of the sensor and walks away from the sensor slowly and steadily for 6 seconds, stops for 3 seconds, and then walks towards the sensor quickly for 6 seconds.

(0,0) Time (s)Po

sitio

n (m

)(0,0) Time (s)

Posi

tion

(m)

(0,0) Time (s)

Posi

tion

(m)

(0,0) Time (s)

Posi

tion

(m)

(0,0) Time (s)

Posi

tion

(m)

6

1. (Work individually) Use a dashed line to sketch your prediction for the position-time graph for this set of motions.

2. (Work together) Compare your predictions. Discuss any differences. Don’t make any changes to your prediction. Once the group agrees on a prediction, quickly sketch it on the white board.

3. (As a class) Compare the computer results with your group’s prediction. Make a sketch of the computer’s results. Explain any important differences between your personal prediction and the results.

Posi

tion

(m)

0

1

2

3

4

0 3 6 9 12 15Time (s)

7

C: Graph MatchingNow for the reverse! To the right is a position-time graph and your challenge is to determine the set of motions which created it.

1. (Work individually) Carefully study the graph above and write down a list of instructions that could describe to someone how to move like the motion in this graph. Use words like fast, slow, towards, away, steady, and standing still. If there are any helpful quantities you can determine, include them.

2. (Work together) Share the set of instructions each member has produced. Do not make any changes to your own instructions. Put together a best attempt from the group to describe this motion. Write up your instructions on the whiteboard to share with the class.

3. (As a class) Observe the results from the computer. Explain any important differences between your predictions and the ones which worked for our “walker”.

D: Exciting CurvesFinally, let’s consider the graph below which curves!

1. Describe how to move to produce this graph – make your best attempt:

2. What is the general difference between motion that produces straight-line position-time graphs and motion that produces curved-line position-time graphs?

Position (m)

(0,0) Time (s)

0 3 6 9 12 15Time (s)

Position (m)

0 1 2 3 4

8

Lesson #3 – Defining Velocity

Pre-Lesson Notes

Unit: Kinematics/Motion Date: Concept: Velocity

Definitions:

Average Speed

Average Velocity

Uniform Motion

Non-uniform Motion

Position

Displacement

Distance

Equations:

Details / Tricks

9

SPH3U: Defining Velocity

To help us describe motion carefully we have been measuring positions at different moments in time. Now we will put this together and come up with an important new physics idea.

A: Changes in Position - DisplacementThe position of an object changes when it moves from one place to another. If we imagine it moving along a straight line that we call the x-axis, we can note its two positions with the symbols d1, for the initial position and d2, for the final position.

1. What is the position of d1 and d2 relative to the origin? Don’t forget the sign convention and units!d1 = d2 =

2. Did the object move in the positive or negative direction? How far is the end position from the starting position? Use a ruler and draw an arrow (just above the line) from the point d1 to d2 to represent this change.

The change in position of an object is called its displacement. The displacement can be found from the quantity: d2 - d1. The Greek letter delta () means “change in” and always describes a final value minus an initial value. We will notate the change in position, or displacement, as d (“delta d”). The displacement can be represented graphically by an arrow, called the displacement vector, pointing from the initial to the final position. Any quantity in physics that includes a direction is a vector.

3. Calculate the displacement of the object in the example above. Consider your answers to question A#2. What is the interpretation of the number part of the result of your calculation? What is the interpretation of the sign of the result?

d = d2 - d1 =

4. Displacement is a vector quantity. Is position a vector quantity? Explain.

5. Calculate the displacement for the following example. Draw a displacement vector that represents the change in position.


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d2

d1

-4 -3 -2 -1 0 1 2 3 4centimeters

| | | | | | | | |

-4 -3 -2 -1 0 1 2 3 4cm

| | | | | | | | |

d1

d2

+ x-axis

10

B: Changes in Position and TimeIn a previous investigation, we have compared the displacement of the physics buggy with the amount of time taken. These two quantities can create an important ratio.

In uniform motion, the velocity of an object is a quantity that describes the displacement that occurs in one unit of time.

1. Write an algebraic equation for the velocity of a uniformly moving object in terms of v, d, d, t and t. (Note: some of these quantities may not be necessary.)

2. Suppose an object moves with uniform motion from d1 = 76 cm to d2 = 13 cm in 0.7 seconds. What is its velocity? Provide an interpretation for the sign of the result.

In physics, there is an important distinction between velocity and speed. Velocity includes a direction while speed does not. Velocity can be positive or negative, speed is always positive. For uniform motion only, the speed is the magnitude (the number part) of the velocity: speed = |velocity|. There is also a similar distinction between displacement and distance. Displacement includes a direction while distance does not. A displacement can be positive or negative, while distance is always positive. For uniform motion only, the distance is the magnitude of the displacement: distance = |displacement|.

C: Velocity and Position-Time GraphsYour last challenge is to find the velocity of a person from a position-time graph.

1. Explain how finding the velocity is different from simply finding the speed.

2. How can you determine the direction in which the person is moving? If you can, give two methods.

3. Determine the velocity of the person between:a) 0 and 6 seconds:

b) 6 and 9 seconds:

c) 9 and 15 seconds:

Posi

tion

(m)

0

1

2

3

4

0 3 6 9 12 15Time (s)

11

Average velocity Homework 1. On your bike, you go 21 km/h [E] for 1 hour, stop for half an hour, then go 10 km/h for 1.5 hours in the

same direction. Determine:a. distance traveled.

b. displacement.

c. average speed.

d. average velocity.

2. A car travels [S] at 60.0 km/h for 5.0 h and then 90.0 km [N] at 72.0 km/h. Determine the:a. distance traveled (what the odometer would show).

b. overall displacement; that is, how far are you from your starting point? (Be careful here: there are two displacements that you must “add”, and one of them is negative.)

c. average speed.

d. average velocity.

Answers: 36 km, 36 km [E], 12 km/h, 12 km/h [E]; 390 km, 210 km [S], 62.4 km/h, 33.6 km/h [S]

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Lesson #4 – Velocity-Time (VT) Graphs

Pre-Lesson Notes

Unit: Kinematics/Motion Date: Concept: Velocity-Time Graphs

Explanation:

Equations:

Details / Tricks

SPH3U: Velocity-Time Graphs

We have had a careful introduction to the idea of velocity. Now it’s time to look at its graphical representation.

A: The Velocity-Time GraphA velocity-time graph uses a sign convention to indicate the direction of motion. We will make some predictions and investigate the results using the motion sensor. Remember that the positive direction is away from the face of the sensor.

1. (Work individually) Sketch your prediction for the velocity-time graph that corresponds to each situation described in the chart below. Use a dashed line for your prediction.

Walking slowly away from the sensor at a steady rate.

Walking quickly away from the sensor at a steady rate. Time (s)

-

Vel

ocity

(m

/s)

+

Time (s)

-

Vel

ocity

(m

/s)

+


0 1 2 3 4 5

13

Start 4 m away and walk slowly towards the sensor at a steady rate

Start 4 m away and walk quickly towards the sensor at a steady rate.

Start 2 m away and walk slowly towards the sensor.

Standing still, about 2 m from the sensor

2. (Work together) Compare your predictions with your group members and discuss any differences. Don’t worry about making changes. On a white board, quickly sketch your six predictions large enough so the whole class can see.

7. (As a class) The computer will display its results for each situation. Draw the results with a solid line on the graphs above. Remember that we want to smooth out the bumps and jiggles from the data.

8. (Work together) Explain to your group members any important differences between your predictions and the results. Record your explanations here.

9. What is the key difference between the graphs made by walking slowly and the ones made by walking quickly?

10. How are the velocity-time graphs different for motion away and motion toward the sensor?

11. What difference does the starting position have on the general shape of a v-t graph?

12. Using the whiteboard, make a sketch of what the corresponding DT graphs would be for the 6 VT graphs above.

Time (s)

-

Vel

ocity

(m

/s)

+

Time (s)

-

Vel

ocity

(m

/s)

+

Time (s)

-

Vel

ocity

(m

/s)

+

Time (s)

-

Vel

ocity

(m

/s)

+

14

B: Prediction Time!A person walks away from the sensor slowly and steadily for 6 seconds, stands still for 6 seconds, and walks towards the sensor steadily about twice as fast as before.

1. (Work individually) Use a dashed line to draw your prediction for the shape of the velocity-time graph for the motion described above.

2. (Work together) Compare your predictions with your group and see if you can all agree. Quickly sketch the group prediction on the whiteboard.

3. (As a class) Compare the computer results with your group’s prediction. Explain to your group members any important differences between your personal prediction and the results. Record your explanations here.

0 3 6 9 12 15Time (s)

Vel

ocity

(m/s

)-1

0 +1

15

SPH3U: Velocity Problem Solving

Now that you have learned a bit about velocity, it’s time to learn a bit about problem solving.

A: How to Write a Clear SolutionWriting a mathematical solution is like writing an essay: your purpose must be clear, your facts must be straight, the logic of your argument must be carefully shown, and your final conclusion definitively stated. You should always write with an interested reader in mind – someone who does not know any physics, but is reasonably good at math. Consider the following solution:

Givens:

d1 = 60 m d2 = 80 m

Required: t = ?

Approach: v = d/t

Solution: Solve for time:t = d/v

Find the displacement:d = d2 - d1 = 80 m – 60 m = 20 m

Calculate the time:t = d/v = (20 m)/(10.7 m/s) = 1.86 s

Paraphrase: The runner took 1.9 s to the distance.

1. Take a few moments to study the solution presented above. Carefully describe what is happening in this problem.

2. What does the series of dots illustrate?

3. Pretend you are writing a problem for a physics textbook. Be creative! Write the original problem for the solution above. Write your problem on a whiteboard to share a bit later.

4. Based on what you see in the solution, describe what is involved in each step.

+ x-axisd2

d1

v = 10.7 m/s∆t = ?


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16

Givens:

Required:

Approach:

Solution:

Paraphrase:

The format for writing this solution is called “GRASP”. You should show all these steps (but you may leave out the labels!) whenever you are carefully writing a solution to a problem.

B: Units and ConversionsWhen we say that something is 3 m long, what do we really mean?

1. “3 metres” or “3 m” is a shorthand way of writing a mathematical calculation. You may not have thought about this before, but there is a mathematical operation (+, -, , ) between the “3” and the “m”. Which one is it? Explain.

Physics uses a standard set of units, called S. I. units, which are not always the ones used in day-to-day life. The S. I. units for distance and time are metres (m) and seconds (s). It is an important skill to be able to change between commonly used units and S.I. units. Do this using a conversion ratio.

2. A student measures a weight to be 0.454 kg. He does a conversion calculation and finds a result of 1.00 lbs. He places a 0.454 kg mass on one side of a balance scale and a 1.00 lb mass on the other side. What will happen to the balance when it is released? Explain what this tells us physically about the two quantities 0.454 kg and 1.00 lbs.

3. There is one number we can multiply a measurement by without changing the size of the physical quantity it represents. What is that number?

17

The process of conversion between two sets of units leaves the physical quantity unchanged – the number and unit parts of the measurement will both change, but the result is always the same physical quantity, just described in a different way. To make sure we don’t change the actual physical quantity when converting, we only ever multiply the measurement by “1”. This means the conversion ratio must always equal “1”. Consider this example, given that 1 kg = 2.203 lbs:

4 kg = 4 kg x ( 2. 203 lbs

1 . 00 kg )= 8.812 lbs

The ratio in the brackets is the conversion ratio. Note that the numerator and denominator are equal, making the ratio equal to “1”. It is often helpful to complete your conversions in the Givens section of your solutions.

4. Describe a handy way of remembering which quantity to put on the top and bottom of the conversion ratio. Where does the unit that you want to get rid of go? A hint comes from the markings and units in the example above.

– check with your teacher before proceeding –

C: Time for Some Problems1. A pitcher throws a fastball with a horizontal speed of 160 km/h. How long does the ball take to reach home

plate 18.4 m away? Solve using GRASP. Hint: make your conversions at the givens step.

2. Usain Bolt ran the 200 m sprint at the 2012 Olympics in London in 19.32 s. Assuming he was moving uniformly, what is his speed in km/h during the race?

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Speed Poster Assignment

Conduct some brief research into the speed of an interesting, unusual thing. Create an attractive poster on an 8½”x14” page that:

has a catchy title. dramatically illustrates the interesting moving thing (your own drawing or a found image.) gives exciting speed facts in units of m/s, km/h and BCI/s (note 1 BCI = 120 m.) shows how long it would take to cross (a) 1 BCI, in seconds, and (b) the time it would take in seconds

and in another appropriate* time unit, to cross your choice of either Waterloo Region or Ontario or Canada or the earth’s circumference (*appropriate here means the number is sensibly sized, or close to 1. E.g. you could say that you are about 500,000 seconds old, but it would be more appropriate to state your age as 16 years old.)

shows the conversion calculations neatly at the bottom or sidebar. shows final answers stated clearly without using a ridiculously large number of significant digits. shows the source for the information that your teacher can check if curious.

Marking – Total = 10 marks0 1 2 3

Interest Really not very interesting

Kind of unusual Very interesting / novel

Appearance No illustration, sloppy layout, title unclear

basic illustration, layout hard to follow, messy, disorganized

Good illustration, tidy layout, easy to read, prominent title

Attractive illustration, eye-catching layout and title

Speed Facts Non-existent or quite wrong

Facts presented but with errors in notation or units

Facts clearly presented with correct units

Conversion Calculations

Not present or contains fundamental errors

Present but sloppy or missing important detail.

Correct but may contain minor errors or omissions.

Clearly and accurately presented, units carefully used

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Lesson #6 – Non-Uniform Motion

Pre-Lesson Notes

Unit: Kinematics/Motion Date: Concept: Non-Uniform Motion

Definitions:Uniform Motion

Non-uniform motion

Acceleration

Explanation:

Equations:

Details / Tricks

SPH3U: Non-Uniform Motion

Part A: Changing Velocity


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A ball rolls from point A to point D on the tracks shown below. The length of each track and the time the ball takes to roll across that section of track is indicated.

Consider the following statement made by a student: “If I want the velocity of the ball on a segment of track, I just take that d and divide by that t. Velocity is just d/t.”

Segment AB1. Do you agree with the statement when it is applied to this segment of track? Explain.

2. Calculate d/t for this segment.

3. Decide whether d/t represents the velocity of the ball at the start of the segment, at the end of the segment or both or somewhere else. Explain.

Segment BC1. Do you agree with the statement when it is applied to this segment of track? Explain.



Segment CD1. Do you agree with the statement when it is applied to this segment of track? Explain.



This example shows us that there is a problem with finding the velocity of an object that is moving non-uniformly. This turns out to be quite a serious problem!

B: Tangents and Velocity

t = 0.75s, d = 2.0 mt = 1.00 s, d = 2.0 m

t = 1.50 s, d = 2.0 mA B

C D

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If a graph is curved, it does not rise by the same amount for each unit of run. If we want to find out how much the graph is rising per unit of run at a particular point (instant), we must look at a small interval that contains the point. If the interval is small enough, the graph behaves much like a straight-line graph. We can then find the slope just as with a straight-line graph: the rise divided by the run.

We interpret the slope as the rise per unit run the graph would have if the graph did not curve anymore, but continued as a straight line. This imaginary straight line is called the tangent to the graph at the point of interest. We then extend this straight line in both directions to form the tangent.

The slope of a position-time graph at an instant gives the instantaneous velocity at that instant.

Consider the graph shown to the right. Always calculate the slope of a curved graph by drawing the tangent, extending it a great distance on both sides of the point of interest, and choosing two points on the tangent that are far apart.

1. Calculate the slope of the tangent that is already drawn for you.

2. Draw a new tangent to the graph at 0.3 s. Determine the instantaneous velocity at 0.3 seconds.

3. Estimate the speed at the beginning, and at the end. Also, where does v = 0?

4. Draw a tangent to your graph at 0.35 s and use it to determine the instantaneous velocity there.

5. List your calculated velocities in order now, at: the beginning, 0.3 s, 0.35 s, 0.425, 0.5 s, and at the end. Consider these as you write a description of the motion, from beginning to end, in one sentence.

Position (cm)

0 10 20 30 40 50 60 70 80 90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time (s)

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Homework: Graph Conversion

Each row in the chart below shows four representations of one motion. Sometimes “x” is used instead of the letter “d” for position. The motion diagram is a dot pattern. If the object remains at rest, simply “pile up” the dots. If it changes direction, use a separate line just above or below the first. Consider the example shown here. Notice that motion diagram origins are marked by a small vertical line and a small arrow shows the starting direction.

Motion Diagram

(Slow to right, stopped, fast to right)

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SPH3U: The Idea of Acceleration

A: The Idea of Acceleration

Interpretations are powerful tools for making calculations. Please answer the following questions by thinking and explaining your reasoning to your partners, rather than by plugging into equations. Consider the situation described below:

A car was traveling with a constant velocity 20 km/h. The driver presses the gas pedal and the car begins to speed up at a steady rate. The driver notices that it takes 3 seconds to speed up from 20 km/h to 50 km/h.

1. How fast is the car going 2 seconds after starting to speed up? Explain.

2. How much time does it take to reach a speed of 75 km/h? Explain.

3. a) A student who is studying this motion subtracts 50 – 20, obtaining 30. How would you interpret the number 30? What are its units?

b) Next, the student divides 30 by 3 to get 10. How would you interpret the number 10? (Hint: don’t use the word acceleration, instead explain what the 10 describes a change in. What are the units?)

B: Watch Your Speed!Shown below are a series of images of a speedometer in a car showing speeds in km/h. Along with each is a clock showing the time (hh:mm:ss). Use these to answer the questions regarding the car’s motion.

1. What type of motion is shown on a speedometer (constant or changing?) Explain.

2. Is the car speeding up or slowing down? Is the change in speed steady? Explain how you can tell.

3. Explain how you could find the acceleration of the car. Calculate this value and write the units as (km/h)/s.


Com / Know / Th / App: 0 1 2 3 4 5


Com / Know / Th / App: 0 1 2 3 4 5


0 1 2 3 4 5

3:50:033:50:023:50:01

600

50

4030

20

10

600

50

4030

20

10

600

50

4030

20

10

3:50:00

600

50

4030

20

10

24

4. A student exclaims, “Why are there two units for time, hours and seconds? This is strange!” Explain to the student the significance of the hours unit and the seconds unit. The brackets provide a hint.

C: Interpreting Motion GraphsAt right is the velocity versus time graph for a particular object.

1. What does the single point A indicate about the object?

2. Give an interpretation of the length labelled c.

3. Give an interpretation of the length labelled d.

4. Give an interpretation of the ratio d/c. How is this related to our discussion in part A?

5. Calculate the ratio d/c including units. Write the units in a similar way to question B#3

6. Use your grade 8 fraction skills to show how to simplify the units of (m /s )/ s or

ms÷s

Vel

ocity

(m/s)

d

c

B

A

0 19876543210

15

10

5

Time (seconds)

25

SPH3U: Calculating Acceleration

A: Defining Acceleration

The number calculated for the slope of the graph in part A of last class’ investigation is called the acceleration. The motions shown in parts A, B and C of that investigation all have the characteristic that the velocity of the object changed by the same amount in equal time intervals. When an object’s motion has this characteristic, we say that the object has constant acceleration. In this case, the total change in velocity is shared equally by all equal time intervals. We can therefore interpret the number Δv/t as the change in velocity occurring in each unit of time. The number, Δv/t, is called the acceleration and is represented by the symbol, a.

a = Δv/t, if acceleration is constant

In Gr. 11 physics, we only deal with situations in which the acceleration is constant. Acceleration can mean speeding up, slowing down, or a change in on object’s direction - any change in velocity qualifies!

1. We mentioned earlier that the “Δ” symbol is a short form. In this case, explain carefully what Δv represents using both words and symbols.

B: Problem Solving1. Suppose an object is speeding up with constant acceleration. At t = 10 s, the object has velocity 4.6 m/s. At t

= 90 seconds, the velocity is 8.2 m/s.

a) What is the acceleration? (answer: a = 0.045 m/s2)

b) What is the velocity at t = 15 minutes? (answer: v = 44.65 m/s ~ 45 m/s with 2 sig.dig.s)Hint: v1 = 4.6 m/s and t1 = 10 s; then use the given t2 converted to seconds.)

In the previous example, if you did your work carefully you should have found units of m/s2 for the acceleration. It is important to understand that the two seconds in m/s/s (m/s2 is shorthand) play different roles. The second in m/s is just part of the unit for velocity (like hour in km/h). The other second is the unit of time we use when telling how much the velocity changes in one unit of time.


0 1 2 3 4 5

26

C: Making Sense of AlgebraConsider the following conversation among three physics students who are trying to do their homework.

Alan: The most efficient way to solve physics problems is to find the right equation and plug in the right quantities.

Beatriz: It’s no use being fast if you end up being wrong. The best way to solve a physics problem is to think it through in a way that makes sense. Then, if you still have to do any equations after that, you can check that they agree with your thinking.

Carly: It’s no use thinking if it doesn’t get you anywhere. You shouldn’t expect physics problems to make sense all the time; sometimes, you need to just use the equations.

Which student do you agree with the most? Explain your thinking.

D: A Few Problems1. Hit the Gas! A car moving with a constant acceleration has a velocity of 5 km/h at t = 0 and a velocity of 30

km/h at t = 5 seconds. Complete all the following questions about this motion.

a) Draw a motion diagram (dot pattern) for the problem and attach the known information.

b) Sketch the velocity-time graph for this situation.

c) Draw a velocity vector for v1 and v2. Use a scale of 1 cm on the page represents 5 km/h of speed. How should their sizes compare?

e) Find the velocity at t = 4 seconds. Explain your reasoning. Do not use algebra!

Time (s)

-

Vel

ocity

(m

/s)

+

27

2. The Rattlesnake Strike The head of a rattlesnake can accelerate at 50 m/s2 in striking a victim. If a car could do as well, how long would it take to reach a speed of 100 km/h from rest?

a) Draw a motion diagram for the problem and attach the known information.

b) Sketch the velocity-time graph

c) Complete the solution using the GRASP steps. (Hint: convert km/h to m/s 1st; answer: t = 0.55s)

3. The Rocket A model rocket is initially at rest, is launched with an acceleration of (50 km/h)/s, and travels for an interval of time, t.

a) Draw a motion diagram for the problem and attach the known information

b) Sketch the velocity-time graph

c) The rocket travels for a time interval of 7.5 s. What is its final velocity? (answer: v2 = 375 km/h)

Time (s)

-

Vel

ocity

(m

/s)

+

Time (s)

-

Vel

ocity

(m

/s)

+

28

SPH3U: Speeding Up or Slowing Down?

There is one mystery concerning acceleration remaining to be solved. Our definition of acceleration, v/t, allows the result to be either positive or negative, but what does that mean? Today we will get to the bottom of this.

Part A: The Sign of the AccelerationConsider this situation: A bicycle accelerates uniformly from a velocity of -3.0 m/s to a velocity of -7.0 m/s. Note that it is always moving in the negative direction, or west, for example. This change in velocity takes 10 s.

1. Without making the acceleration calculation, explain what the sign of the bicycle’s acceleration will be.

2. (work individually) The sign of the acceleration is not easy to interpret. Consider the four velocity-time graphs below. For each graph indicate: the sign of the velocity, whether its magnitude (speed) is getting larger or smaller, the sign of the acceleration, whether the object is speeding up or slowing down. Try each row individually and then compare your results with your group members.

Graph Sign of Velocity Magnitude ofVelocity

Sign of Acceleration

SU or SD?

1

2

3

4


0 1 2 3 4 5

Time

-

V

eloc

ity

+

Time

-

Vel

ocity

+

Time

-

V

eloc

ity

+

Time

-

Vel

ocity

+

29

3. Which graph corresponds to the bicycle example we started with? Explain.

4. Which quantity from the chart tells us the direction of motion of the object? Explain.

Now let’s try to interpret the sign of the acceleration carefully. Acceleration is a vector quantity, so the sign indicates a direction. This is not the direction of the object’s motion! To understand what it is the direction of, we must do some more work.

5. Can we convincingly state that if the acceleration is positive the object is speeding up? What about a negative acceleration and slowing down?

6. What conditions for the acceleration and velocity must be true for an object to be speeding up? To be slowing down?

Always compare the magnitudes of the velocities, the speeds, using the terms faster or slower. Describe the motion of accelerating objects as speeding up or slowing down and state whether it is moving in the positive of negative direction. Other ways of describing velocity often lead to ambiguity and trouble! Never use the d-word, deceleration - yikes! Note that we will always assume the acceleration is uniform unless there is a good reason to believe otherwise.

Part B: Speeding Up and Slowing Down on Position-Time GraphsOur final step in understanding this business of speeding up and slowing down is their representation in a position-time graph. Consider the position-time graph shown below which represents the motion of a cart with a fan attached to it. The fan gives a steady, gentle push which changes the velocity of the cart.

1. Use the two tangents to the graph to help you to explain whether the cart is speeding up or slowing down in this example.

2. What are the directions of the velocity and acceleration? Explain how you can tell.

Position

Time

30

Use the “Moving Man” simulation, found on the website below, to test your graphing predictions in the following chart. http://phet.colorado.edu/en/simulation/moving-man

3. (work individually) For each example (each column), use a dashed line to sketch your prediction for the position-, velocity-, and acceleration-time graphs that will be produced. Complete each example individually and then compare your predictions with the group. Note: It may be easiest to start with the v-t graph and the acceleration-time graph is new! Let’s agree that [Right] is (+).

1 2 3 4Description d1 = 0

v1 = 0a = +2 m/s2 (Continue like this until sudden crash.)

Starting at home, at an initial velocity of 4 m/s [L], you slow down at 2 m/s2 until you stop, & continue in opp. dir.

d1 = - 5 m v1 = + 6 m/sa = - 3 m/s2

d1 = 8 m [R]v1 = 4 m/s [L]a = 2 m/s2 [R]

Position graph

Velocitygraph

Accelerationgraph

Slowing down or speeding up?

4. Use your chart to help explain what the sign of the acceleration represents.

5. What happens when acceleration and velocity have opposite signs (that is, when they are in the opposite direction?)

31

http://phet.colorado.edu/en/simulation/moving-man

Homework: SU/SD 1. Answer the following questions based on the graph. Provide a brief explanation how you could tell.

a) At what times, if any, does the object have a positive acceleration and a negative velocity?

b) At what times, if any does the object have a negative acceleration and a positive velocity?

c) At what times, if any, was the acceleration zero?

d) At what times, if any, was the object speeding up?

e) At what times, if any, was the object slowing down?

f) At what times, if any, did the object sit still for an extended period of time?

g) Overall, is the motion in the graph an example of uniform or nonuniform acceleration?

2. Answer the following questions based on the graph. Provide a brief explanation how you could tell. At which of the lettered points on the graph below: a) is the motion slowest?

b) is the object speeding up?

c) is the object slowing down?

d) is the object turning around?

Vel

ocity

(m/s

)

-20 -10 0 10 20

0 10987654321

Time (seconds)

32

3. A car’s velocity changes from +100 km/h to +10 km/h in 0.1 hours. Is the acceleration positive or negative? Find the acceleration.

4. At the beginning of a half-hour time period, a snail is moving at -3.0 mm/s. The snail then slows down, turns around and starts heading back in the opposite direction at +1.0 mm/s. Is the acceleration positive or negative? Find the acceleration.

5. A car’s velocity changes from 100 km/h [East] to +120 km/h [East] in 5 seconds. Is the acceleration positive or negative? Find the acceleration in m/s2 and km/h2.

6. A hula-hoop is thrown North at 4.0 m/s, but due to a back-spin, it slows down, stops for an instant, and speeds up back to the thrower who catches it going 2.5 m/s [South] 3.0 seconds later. Find the acceleration. If the original direction, North, is positive, state the acceleration as + or – first, then state it with no sign but with the appropriate direction.

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SPH3U: Area and AveragesA graph is more than just a line or a curve. We will discover a very handy new property of graphs which has been right under our noses (and graphs) all this time!

A: Looking Under the GraphA car drives along a straight road at 20 m/s. It is straight-forward to find the displacement of the car between 5 to 20 seconds. But instead, let’s looks at the velocity-time graphs and find another way to represent this displacement.

1. Explain how to calculate the displacement of the car, in a familiar way, like we’ve done many times before.

Now let’s map out that calculation visually - on the graph. There are two important quantities to represent – the velocity and the time interval.

2. Draw a dark horizontal line along the time axis of the graph that spans the time interval during which the car was moving.

3. Draw two vertical lines (parallel to the velocity axis) starting from the time axis and going up to the 20 m/s velocity line of the graph.

4. You have now constructed a shape which we can imagine located between the time axis and the line of the graph. What shape is formed? Shade in the shape on the graph. What characteristic of the shape could you easily calculate since you know the “length” of its two sides?

5. Perform this calculation for the area shaded in your graph. Be sure to show the proper physics units for each quantity used. What quantity does the final result represent?

You can imagine the line along the velocity axis travelling horizontally along the time interval and “sweeping out” an area corresponding to the car’s displacement. In our example above, the larger the time interval, the greater the area that is swept out.

The area under a graph of velocity verses time for an interval of motion gives the displacement during that interval. Both velocity and displacement are vector quantities and can be positive or negative depending on their directions. According to our usual sign convention, areas above the time axis are positive and areas below the time axis are negative.


0 1 2 3 4 5

Velocity (m/s

) 0 20

0 10 20 Time (s)

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B: Kinky GraphsHere’s a funny-looking graph. It has a kink or corner in it. What’s happening here?

1. What characteristic of the graph is steady before the kink, steady after the kink, but changes right at the kink? What has happened to the motion of the object?

At exactly t = 0.5 seconds we cannot tell what the slope is – it is experiencing an abrupt change.

2. Sketch a velocity-time graph for this same motion.

To indicate a sudden change on a physics graph like the one above, use a dashed vertical line. This indicates that you understand there is a sudden change, but you also understand that you cannot have a truly vertical line.

3. What would a vertical line segment on a v-t graph mean? Is this physically reasonable? Explain.

4. Explain how to calculate the area under the v-t graph above. Find this result. Explain how to use the position-time graph to confirm your result.

5. We can perform a new type of calculation by dividing the total area we found by the total time interval. Carry out this calculation and carefully show the units.

6. What type of velocity did you find from the previous calculation? How does it compare with the values in the v-t graph? Overall (during the entire time interval), is motion of the object uniform of nonuniform?

Part C: Average Velocity

Position (cm)

0 10 20 30 40 50 60 70 80 90 100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Time (s)

0 0.5 1.0 Time (s)

Vel

ocity

(cm

/s)

100

35

Earlier in this unit, we noted that the ratio, d/t, has no simple interpretation if the motion of the object is non-uniform. Since the velocity is changing during the time interval, this ratio gives an average velocity for that time interval. One way to think about it is this: d/t is the velocity the object would have if it moved with uniform motion through the same displacement in the same amount of time.

1. On the kinky d-t graph in Part B, use the interpretation above to help you draw a single line (uniform motion) from 0,0 to the final position point. Show that its slope equals the average velocity for that time interval. Show your work on the graph.

2. Use the same process to find the average velocity of the object in the graph to the right from t = 2 to t = 7 seconds. Show your work below.

3. Explain how, in two simple steps, you would find the average velocity for any interval on any position-time graph.

Position (cm)

0 10 20 30 40 50 60 70 80 90 100

0 1 2 3 4 5 6 7 8 9 10Time (s)

36

SPH3U: How Far? (The Displacement Problem)

In the previous section, you discovered how to get displacement from motion graphs. It would also be nice to be able to calculate the displacement without using a graph. Let’s employ the graphs one more time to help us construct another way to determine the displacement, as well as other aspects of motion.

A: Some Discrepancies Arising from Changing Speed

Imagine a car that speeds up from 10 m/s to 20 m/s in 5 seconds. Remember: the area undera line on a v-t graph can tell you how far theobject in motion went.

1. So how can you figure out the area under the line shown here? Draw dotted lines to show a triangleand a rectangle under the line.

A□ = A∆ =

distance (total) = A□ + A∆ =

2. Remember the simplified equation d = v x t ? Try it for the above situation. You can’t get the same answer as you did with the area calculation, can you?

3. Suggest a reason why the simple equation d = v x t doesn’t work.

4. What would the “d” be if the speed was 10 m/s the whole time?

5. What would the “d” be if the speed was 20 m/s the whole time?

6. Did you notice that the actual “d” is the ________________ of these two values?

If something changes its velocity (in a consistent way – so that the v-t line is straight), we can use the average of the initial and final velocity to figure out the total displacement. Also, it would be nice to not always have to use a graph!

Consider our example again: v1 = 10 m/s, and v2 = 20 m/s, and ∆t = 5 s. If we take the average speed of 15 m/s and multiply that by the time, we get d = 15 m/s x 5s = 75 m. It works!

How did we get the average speed? Easy: (10 + 20) /2 = 15; or (v1 + v2)/2

So we just came up with a new equation! Displacement equals average speed x time!

7. Try a few examples:a) A cycle going 5 m/s speeds up to 10 m/s in 2 s.

How far did it go?


0 1 2 3 4 5

t (s)

v (m/s)

10

20

50

∆t2

Δd = (v1+v2)

37

b) A car starts at rest and accelerates to 20 m/s (that’s 72 km/h) in 10 s. Calculate its displacement.

c) You walk at 1.5 m/s for one hour. How far do you go?

B: Relating Graphs to EquationsIn some situations when you’re asked to calculate how far something went, the starting speed (10 m/s) or the final speed (20 m/s) or the time (5 s) are not always all given. Since there are different situations with different variables given to us, we need some general equations to figure out the required answers. Look at this v-t graph for a more general situation:

∆d =

2. So for the general equation we get:

This is the general equation for displacement which can tell us where something is at any time (Δd) if we know its starting speed (v1) and its acceleration (a).

– call your teacher over now to check your equation and your progress –

t (s)

v (m/s)

v1

v2

t2t1

1. Now, instead of using the specific numbers on the graph like before, we’ll have to use the general variables as shown on the graph.

A□ = A∆ =

∆d = A□ + A∆ =

But remember, a = slope (on a v-t) so a = ∆v / ∆t, or ∆v = a x ∆t.

So we can replace the ∆v in the equation above with (a ∆t) instead. Do that here, then simplify it:

∆t

∆v

Δd =

38

3. Let’s try it out for the car from before. Let’s say v1 = 10 m/s, and the car is accelerating at 2 m/s2 (but you could have figured that out using the slope, right?!) What is the displacement after 5 seconds?

4. A car starts out with a speed of 10 m/s, but it is accelerating at 12 m/s2; how far does it go after 5 s?(Answer: 200 m)

5. How far does a car go which starts out at rest and accelerates at 12 m/s2 for 5 s? (Answer: 150 m)

6. What would the equation look like if a = 0? (Does it look familiar?!)

It might also happen that t is unknown, and you need to solve for another variable, like v2. One more motion equation is perfect for this:

7. A car going 10 m/s accelerates at 4 m/s2 over a displacement 50 m. What is its final velocity?

Displacement Problem Homework

1. Zero S -- The “Zero S” is an electric motorcycle which can go from 0 to 100 km/h (or 28 m/s) in 4.8 seconds. What distance is required to get this bike up to speed? (Don’t forget to use a clear list of variables with all your solutions!)

2. Stopping a Muon -- A muon (a subatomic particle) moving in a straight line enters a region with a speed of 5.00 x 106 m/s and then is slowed down at the rate of 1.25 x 1014 m/s2. How far does the muon take to stop?

3. Taking Off -- A jumbo jet must reach a speed of 360 km/h on the runway for takeoff. What is the smallest constant acceleration needed to takeoff from a 1.80 km runway? Give your answer in m/s2

4. Shuffleboard Disk -- A shuffleboard disk is accelerated at a constant rate from rest to a speed of 6.0 m/s over a 1.8 m distance by a player using a cue. At this point the disk loses contact with the cue and slows at a constant rate of 2.5 m/s2 until it stops. What total distance does the disk travel?

SPH3U: The BIG Five Recorder: __________________Manager: __________________Speaker: __________________

0 1 2 3 4 5

Answers: (1) 67 m; (2) 0.10 m; (3) 2.78 m/s2; (4) 9 m

v22 = v1

2 + 2a∆d

39

Last class we found three equations to help describe uniformly accelerated motion. A bit more work along those lines would allow us to find two more giving us a complete set of equations for the five kinematic quantities (variables).

Part A: The BIG Five – Revealed! Here are the BIG five equations for uniformly accelerated motion.

1. Fill in the chart with a check mark () for each variable is found in that equation.

2. How many different variables does each equation have?

3. If you wanted to use the first equation to calculate the acceleration, how many other quantities would you need to know?

4. Define carefully each of the kinematic quantities in the chart below.

5. What condition must hold true (think of our model) for these equations to give reasonable or realistic results?

Part B: As Easy as 3-4-5Solving a problem involving uniformly accelerated motion is as easy as 3-4-5. As soon as you know three quantities, you can always find a fourth using a BIG five! Write your solutions carefully using the GRASP process. The Givens step will help you identify three quantities, the Required step will help you identify the fourth, and the Approach step is where you write your BIG five equation. Use the chart to help you choose a BIG five. Here are some sample situations that the BIG five help describe.

Problem 1A traffic light turns green and an anxious student floors the gas pedal, causing the car to accelerate at 3.4 m/s 2 for a total of 10.0 seconds. We wonder: How far did the car travel in that time (and what’s the big rush anyways?)

40

The BIG Five v1 v2 d a t

v2 = v1 + at

d = v1t + ½at2

d = v2t - ½at2

d = ½(v1 + v2)t

(v2)2 = (v1)2 + 2ad

v1

v2

d

a

t

Draw a motion diagram of the situation, label the two events involved and attach the given information.

Event :

Event :

Sketch a velocity-time graph for the problem

1. Sometimes students get stuck and say, “I am given only two numbers, the acceleration and time. I need three to solve the problem. I’m stuck!” Explain how to help the student.

Complete the remaining GRASP steps and solve this problem.

(answer: d = 170 m)

Problem 2An automobile safety laboratory performs crash tests of vehicles to ensure their safety in high-speed collisions. The engineers set up a head-on crash test for a Smart Car which collides with a solid barrier. The engineers know the car initially travels at 100 km/h and the car crumples 0.78 m during the collision. The engineers have a couple of questions: How much time does the collision take? What was the car’s acceleration during the collision?

Draw a motion diagram of the situation, label the two events Sketch a velocity-time graph for the

Time (s)

-

Vel

ocity

(m

/s)

+

41

involved and attach the given information.

Event :

Event :

problem

Solve this problem here, and include a clear list of given variables.

(answer: a = -4.9 x 102 m/s2 and t = 0.056s)

Problem 3Speed Trap The brakes on your car are capable of slowing down your car at a rate of 5.2 m/s2. You are travelling at 137 km/h when you see a cop with a radar gun pointing right at you! What is the minimum time in which you can get your car under the 100 km/h speed limit?

Solution:

(answer: t = 2.03 s)

Time (s)

-

Vel

ocity

(m

/s)

+

42

SPH3U: Freefall

One of the most important examples of motion is that of falling objects. When an object is tossed or dropped we say that it is in freefall.

Part A: Predicting Falling Motions1. Predict how a rubber ball would fall in as much detail as possible. Be sure to describe if the motion changes

part way through. Explain the reasons for your predictions.

2. Predict how a coffee filter would fall in as much detail as possible. Be sure to describe if the motion changes part way through. Do you think the filter will fall more slowly or more rapidly than the rubber ball? Explain the reasons for your predictions.

3. Now predict how the coffee filter would fall if it is crumpled into a little ball. Be sure to describe if the motion changes part way through. Do you think the crumpled filter will fall more slowly or more rapidly than the rubber ball? Explain the reasons for your predictions.

Get a ball and coffee filter, drop them at the same time and see what happens. Then crumple the filter and try the ball and crumpled filter at the same time.

4. Describe your observations. How did they compare with your predictions? Offer a new explanation if your observations differed from your predictions.

Part B: Describing how Objects Rise and Fall1. Toss the ball straight up a couple of times and then describe how you think it might be moving when it is

moving upwards. Some possibilities include: (a) rising at a constant velocity; (b) rising with an increasing acceleration; (c) rising with a decreasing acceleration; or (4) rising at a constant acceleration. What do you think and why?

2. Describe how you think the ball might be moving when it is moving downwards. Some possibilities include: (a) falling at a constant velocity; (b) falling with an increasing acceleration; (c) falling with a decreasing


0 1 2 3 4 5

Adapted from Workshop Physics Activity Guide: I – Mechanics, Laws, John Wiley & Sons, 2004

43

Time

Position

sketchdetector

ball

Time

Velocity

acceleration; or (d) falling at a constant acceleration. What do you think and why?

3. Do you expect the acceleration when the ball is rising to be different in some way than the acceleration when the ball is falling? Why or why not?

4. What do you think the acceleration will be at the moment when the ball is at its highest point? Why?

Part C: Analyzing the Motion of a Tossed BallAs a class, observe the motion of the tossed ball using the motion detector attached to the ceiling.

1. (as a class) In the sketch of the apparatus indicate the physical location of the origin used for the detector measurements. Use an arrow to indicate which direction is positive.

2. (individually) The ball with travel up towards the detector and back down. Use a dotted lie to sketch a prediction for the position-time graph for this trip.

3. (as a class) Observe the results from the computer. Carefully sketch a position-time graph for the vertical motion. Note that we will use the symbol y for positions along a vertical or y-axis.

4. Label the portion of the graph that represents upwards motion, downwards motion and the highest point in its trip. Indicate in which portions the velocity positive, negative, or equal to zero.

5. Now consider a velocity-time graph for the ball. Indicate on the velocity axis which direction is positive.

44

6. (individually) Use a dotted line to sketch a prediction for the velocity-time graph.

7. (as a class) Sketch a velocity-time graph for the tossed ball based on the computer results.

8. Label the portion of the graph that represents upwards motion, downwards motion and the highest point in its trip. Indicate in which portions the velocity positive, negative, or equal to zero.

9. Do these observations confirm your prediction regarding the nature of the upwards motion? Explain.

10. Do these observations confirm your prediction regarding the nature of the downwards motion? Explain.

11. Is the ball’s acceleration as it rises the same or different from its acceleration as it falls? What do you conclude about the acceleration of a tossed ball?

12. Many people are interested in what happened when the ball “turns around” at the top of its trip. Some students argue that the acceleration at the top is zero; others think not. What do you think happens to the acceleration at this point? Use the v-t graph to help explain.

45

SPH3U: The Acceleration Due to Gravity

Last class we saw that an object in freefall accelerates uniformly due to gravity when air resistance is not a large factor.

A: The Acceleration Experiment: a Bouncing BallPut this lab in your lab duotang. A bouncy ball will be dropped from a height of 2 metres and will be allowed to bounce twice. By viewing a video of the event, we will be able to determine the height at any time during its trip.

Raw data section1. Prediction #1: the motion of a bouncing ball. Make 3 sketches of its motion right now, including both

bounces. We’re going to start at a position of 2 m [up], until the maximum height of the 2nd bounce. Let’s agree now that “up” is positive and down is negative. First draw a d-t graph, second, a v-t graph, and third, an a-t graph, all in one column so you can line-up the times. Don’t be afraid of making mistakes here; you won’t be penalized. (Hint: d starts at +2 and goes to 0 before heading back up; v starts at 0 but increases down so it gets more and more negative…)

2. Prediction #2: What maximum speed would you guess the ball achieves just before it hits the ground?3. Analyze the video to get the positions and the times. Put this data in a chart in your lab duotang.

Don’t forget to follow the lab report guidelines you received in the intro package. For full marks, include 2 or 3 new observation charts in the formal lab write up, even if your raw data is legible. Use graph paper for the graphs and include them before the Analysis Questions section. Make sure they are large, have titles, and that each axis is clearly labeled, including units.

4. Use the position vs. time data to create the actual d-t graph. Circle your points and draw a smooth curve through the points.

5. Calculate the change in position (∆d = d2-d1) for each time interval (∆t = _____ s), and from this the velocity (v = ∆d / ∆t) at each interval. Add columns to the d-t table or make a new table for this data. Remember that v should turn out to be negative at first, since each d2 for the first points are smaller than each d1 as the ball falls.

6. To plot an actual v-t graph, you need t values for each v. For the times, use the mid-point in time for each interval (e.g. if your ∆d occurred from 0.2 to 0.3 s, plot it at a time of 0.25 s.) Make your v-t graph. Circle your points again, and draw lines of best fit for any groups of points that seem to be linear.

7. Calculate the slopes of the straight lines on the v-t graph and put this data into a new table. (Hint: there should be 3 clear slopes, all negative. But depending on how good our data is, we may be able to see some large positive slopes at each bounce too.)

8. “Plot” an acceleration-time (a-t) graph using this data. (Hint: Assume acceleration is constant for each straight-line section, as it appears to be on the v-t graph. So this should appear as a flat, horizontal line, negative, for each of the 3 sections.)

Analysis Questions1. Mark places on your d-t and v-t graph where the ball was stopped. At what times did this occur?2. At what heights did this occur?3. How can you use your graph to estimate how long the ball was in contact with the floor each time?

Estimate these times.4. What is the highest speed the ball achieved? Where and when did this occur?5. Using the 3 v-t slopes (negative accelerations), calculate the average acceleration while the ball was

free-falling (this excludes the bounces.)6. The accepted value for free-fall acceleration is 9.8 m/s2. Calculate your % difference between this value

and your average free-fall acceleration.7. At the acceleration you calculated, how fast should a free-falling ball be going after:

a) 1 s b) 2 s c) 20 s d) one minute?


0 1 2 3 4 5

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8. Calculate the area under the first triangular section of the v-t graph. This should include one big triangle (starting at zero) and one small sliver of a triangle (ending at zero during the bounce.)

9. Calculate a % difference between the area from the previous question (experimental) and the distance the ball should have travelled so far (actual). Hopefully your % diff. is small. What sources of error would lead to any % diff. at all?

10. Compare each of your graph predictions to the experimental results and comment. 11. Calculate the area under each section of the a-t graph. What significance do these values have?

Bonus: According to our understanding of d-t graphs, a flat and horizontal line represents no motion, or stopped. But the ball stopped more than three times during its entire 2 bounce journey. But are there that many flat horizontal sections on your d-t graph? Explain this discrepancy.

47

SPH3U: Graphing Review

This exercise will help you put together many of the ideas we have come across in studying graphical representations of motion. Before completing the table, examine the sample entries and be sure you understand them. Note that some information may be found on both graphs. Now fill in the missing entries in the table below – describe how you find the information.

How to get information from motion graphs

Information sought Position-time graph Velocity-time graph

Where the object is at a particular instant

The object’s velocity at an instant

The object’s acceleration (if constant) Can’t tell Compute the slope

Whether the motion is uniform

Whether the object is speeding upCheck whether the curve is getting

steeper

Whether the object is slowing down

Whether the acceleration is constant

The object’s change in position

The object’s change in velocity

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gr - · web viewwe call the motion of the buggy uniform motion. use your ideas from the...

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