Sept 9: Chapter 2Describing Motion: Kinematics in One Dimension

Warning: former students say kinematics is hard at first………..but by December they say it is easy

First, a question

• You walk at 0.80 m/s through the woods. How long will it take for you to travel 2.444 km?a. Report in seconds.

b. Report in hours.

Manage sig figs.

Managing homework

• If you don’t have 75% of your assignment done…– Give yourself a HW check– Write date, name, and assignment you

missed in “the purple notebook”

• If, two classes later, you haven’t done it, you’ll get one more HW check

Sections of Chapter 2 we will cover…•Reference Frames and Displacement

•Average Velocity

•Instantaneous Velocity

•Acceleration

•Motion at Constant Acceleration

•Solving Problems

•Falling Objects

•Graphical Analysis of Linear Motion

Speed of the bowling ball

• Using only what you have, estimate the speed as I roll it.

• What is the velocity?

• It is 100,000 km/hr! Yes, it is.

• What is the acceleration?

• We need to define our reference for measurement.

2-1 Reference FramesAny measurement of position, distance, or speed must be made with respect to a reference frame.

For example, if you are sitting on a train and someone walks down the aisle, their speed with respect to the train is a few miles per hour, at most. Their speed with respect to the ground is much higher.

2-1 Displacement

We make a distinction between distance and displacement.

Displacement (blue line) is how far the object is from its starting point, regardless of how it got there.

Distance traveled (dashed line) is measured along the actual path taken.

We will use both.

2-1 Displacement

The displacement is written:

Left:

Displacement is positive.

Right:

Displacement is negative.

2-2 Average Speed & Velocity

Speed: how far an object travels in a given time interval

Velocity includes directional information:

(2-1)

Check: *can avg speed be lower than avg velocity?* What is slowest avg speed? Highest? …

check

• Yesterday, this book was at my desk, and today it is here. Calculate average v.

• North America drifts about 1 inch per year.– What is average velocity in SI units?

2-3 Instantaneous Velocity

The instantaneous velocity is like the average velocity, in the limit as the time interval becomes infinitesimally short.

These graphs show (a) constant velocity and (b) varying velocity.

(2-3)

Check: graph my acceleration.

Velocity with go-motion

• Observe this cart motion and estimate the instantaneous velocity in SI units.

• Now check it with go-motion device.

Let’s talk graphs and carts

• A car on a track, x is position from origin

• What is positive, what is negative?

• Now you Plot x vs. t, identify origin (0,0)– X is plotted on vertical but represents horizontal !?!?– What does slope tell us?– What is velocity, what does it look like?– Think about physical motion, and think about

graphical representation of that motion.

• Check: given constant velocity, what is a?

Make graphs from motion

1. bowling ball on ground

2. Fan car on ground

3. Fan car up a sloped table, no motion

4. Bowling ball down sloped table

5. Bowling ball up a sloped table

a

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Take the motion graph challenge on loggerpro

• Loggerpro FILE/OPEN/Physics/graphmatching

• 1b

• 1c

• 1d

• 1e velocity

• 1g velocity

*2-4 AccelerationAcceleration is the rate of change of velocity. (How fast is velocity changing)

2-4 Acceleration

Acceleration, like velocity is a vector, although in one-dimensional motion we only need the sign.

The previous image shows positive acceleration; here is negative acceleration with positive velocity:

2-4 more Acceleration

There is a difference between negative acceleration and deceleration:

Negative acceleration is acceleration in the negative direction as defined by the coordinate system.

Deceleration occurs only the acceleration is opposite in direction to the velocity.

I recommend avoiding the term deceleration in this class.

2-4 Acceleration

The instantaneous acceleration is the average acceleration, in the limit as the time interval becomes infinitesimally short.

(2-5)

It is a snapshot of the acceleration at a givenmoment.Check: graph my acceleration…

Acceleration check

• A car drives 25m/s and then accelerates to 45m/s in 10.0 seconds.

• What is average acceleration?

• In the morning, I drive my car from our garage to the school and park. My maximum speed is 45mph and the trip takes me 10 minutes. What is average acceleration of my car?

We can write the average velocity of an object during a time interval t as

The acceleration, now assumed or known to be constant, is

Special case: Motion at Constant Acceleration (the math that works for us)

What do these all mean?

Check: estimate the a for this motion…

2-5 Motion at Constant Acceleration, deriving the useful equations.

In addition, as the velocity is increasing at a constant rate, we know that the average is:

Combining these last three equations, we find:

(2-8)

(2-9)

Note: I will not expect you to derive these kinematic equations…

2-5 Motion at Constant Acceleration, the kinematic equations

We can also combine these equations so as to eliminate t:

We now have all the equations we need to solve constant-acceleration problems.

(2-10)

(2-11a)

(2-11b)

(2-11c)

(2-11d)

A typical (classic) problem• You push gently on the accelerator of your

brand new car, and the car moves from a stop to 22m/s in 63 m. Find a.

• Draw at beginning and at end = 2 pictures• What do you know at each time?• What equation will use this info and give you a?• Solve symbolically for a and check units.• Now calculate the number and sig figs.• Does your answer make sense?

*“The Big 7” Problem solving steps

1. Draw and label the situation at two times1. Write something for xo, x, to, t, vo, v, and a

2. Decide and show which direction is +

2. Find the correct kinematic equation

3. Delete what is zero

4. Solve algebra for desired unknown (do one unknown at a time)

5. Check the units of the algebraic equation

6. Plug in the numbers and units, calculate.

7. Fix answer for correct sig figs

2-6 Solving Problems (Big 7 are in red)

1. Read the whole problem and make sure you understand it. Then read it again.

2. Decide on the objects under study and what the time interval is.

3. Draw a diagram and choose coordinate axes.

4. Write down the known (given) quantities, and then the unknown ones that you need to find.

5. What physics applies here? Plan an approach to a solution.

2-6 Solving Problems (cont)

6. Which equations relate the known and unknown quantities? Are they valid in this situation? Solve algebraically for the unknown quantities, and check that your result is sensible (correct dimensions).

7. Calculate the solution and round it to the appropriate number of significant figures.

8. Look at the result – is it reasonable? Does it agree with a rough estimate?

9. Check the units again.

Constant acceleration

• It is one number

• It is one value

• It can be zero

• What are examples?

Demo with fan car and GoMotion

• Measure x, v, a

• Calculate a and check it

2-7 Falling Objects

Near the surface of the Earth, all objects experience approximately the same acceleration due to gravity.

This is one of the most common examples of motion with constant acceleration.

We love it.

2-7 Falling Objects

In the absence of air resistance, all objects fall with the same acceleration, although this may be hard to tell by testing in an environment where there is air resistance.

Demo: book and paper, vacuum and feather

• Sketch the set up and show me

• Write the procedure in 4-5 bullet points

• write these answers– What happens to the air in the chamber?– Why do paper and feather fall slowly?– Why does magnet fall quickly?– Why does feather fall slower than paper?– Why do meteorites burn up in the atmosphere?– Can anything fall in air without any friction?

2-7 Falling Objects, the data

The acceleration due to gravity at the Earth’s surface is approximately 9.80 m/s2.

How to solve vertical motion kinematics?

•Same as horizontal!

•And know that a = -g when up is positive

Look at HW problems

• Practice together

• Do example 2-5

• And do p-27 together in class

15 sept: More on graphs

• Plots of x vs. t or v vs. t– Look at 6 cases on next page– Describe motion of each– Derive companion graphs from each– Always do graph down first– Add numbers if it helps– Be sure you can understand units of each

The importa

nt 6 graphs

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Quiz on motion

• A fan car has constant acceleration from 4.0m/s to 1.0 m/s in 2.0 m. – What is a?

– What is tf?

– Plot the motion graphs x, v, and a without numbers

Understanding x, v, and a

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

• To understand motion, we need to understand the graphs of x, v, and a vs. t, and know how those relate to actual motion.

• Check– X,v,a, of no motion, constant position– Xva of constant velocity– Xva of constant acceleration

*Check your “graphs understanding”

• Draw x,v,a vs. t for each motion

• Motions: – fan car push away– Ball toss up– Ball bounce– Falling coffee filter

x

v

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Now make it real

• Write each of these motion graphs on a page you can carry.

• Go into the hall and practice each motion as a table, reach consensus.

• I will quiz you in 5 minutes using Loggerpro and this cart.

• Each group will match the graph on Loggerpro

2-8 Graphical Analysis of Linear Motion

This is a graph of x vs. t for an object moving with constant velocity. The velocity is the slope of the x-t curve.

You’ve done this in math?

Graphical analysis made easy

• How far did this object go in 5s?

• What is the area under this curve?

• The same is true for any shape of v-t curve!

v

t0 5s

6m/sArea under the v-t curve = the distance traveled

2-8 Graphical Analysis of Linear Motion

On the left we have a graph of velocity vs. time for an object with varying velocity; on the right we have the resulting x vs. t curve. The instantaneous velocity is tangent to the curve at each point.

2-8 Graphical Analysis of Linear Motion

The displacement, x, is the area beneath the v vs. t curve.

Don’t need this yet!

Quiz on a classic problem

• I stand on the school roof and drop a rock from 6.0 m. – Draw a picture.– Write in all known and unknown values for t,

y, v, and a.– How long does it take to hit the ground?– How fast is it going when it hits?

Summary of Chapter 2

• Kinematics is the description of how objects move with respect to a defined reference frame.

• Displacement is the change in position of an object.

• Average speed is the distance traveled divided by the time it took; average velocity is the displacement divided by the time.

• Instantaneous velocity is the limit as the time becomes infinitesimally short.

Summary of Chapter 2 cont.• Average acceleration is the change in velocity divided by the time.

• Instantaneous acceleration is the limit as the time interval becomes infinitesimally small.

• The equations of motion for constant acceleration are given in the text; there are four, each one of which requires a different set of quantities.

• Objects falling (or having been projected) near the surface of the Earth experience a gravitational acceleration of 9.80 m/s2.

•We can solve problems using 7 steps…

Practice during single sessions: 16 P 26 and 57

Classics chapter 21. Simple horizontal acceleration; fan car

2. Ball off a cliff, with v0 = 0 (use newton ball to demo)

3. Ball off a cliff, but with initial velocity down

4. Ball thrown up and falls down

5. Car braking to a stop, how far, or time

6. Car into tree, front compresses, what is a?

7. Guy falls into net, decelerates, two parts

8. The 6 graphs of x, v, a vs t

Comments on exam 1

• Physics (and Chem) are often the first classes where some students get grades less than 90%. Was true for me.

• Don’t panic on this test.

• You have seen all the techniques: algebra, vectors, solving problems

• You just need practice.

• You can do this.

Exam 1 post mortem– Mean = average = 89%– Low=60; high = 100

• Review with different pen• Record correct answers• Let me know after class if addition or

correction mistake by me• Take exam and correct any answer for

Friday. Make this legible. Attach sheet to show new work.

Lab 1: velocity and acceleration of a bowling ball

5 min: agree on plan, assign tasks5 min: set up and take a trial run with timers15 min: conduct runs with GoMotion and timers10 min: short present to Dr. F of results. Save data5 min: clean up

• 3 per group.• Share a ball to get set up.• Find or build a ramp.

TC marathon