chapter 2. one-dimensional kinematics -...

Copyright © 2010 Pearson Education, Inc.

Chapter 2. One-Dimensional

Kinematics

Chapter 2.1. Displacement and

Velocity


Mechanics: The Study of Motion


Frame of Reference Describing motion requires a frame of

reference to specify how a object’s position

changes over time. We refer to a frame of

reference with a coordinate system.

The origin indicates

where x=0, y=0, z=0,

t=0, etc. Numeric

values of x, y, z and t

are relative to this

point.


Defining Position

Position indicates your location relative to the

origin of your coordinate system (frame of

reference). Position can be positive or negative.

xi = initial position, xf = final position


Defining Displacement

Left:

Displacement is positive.

Right:

Displacement is negative.


Defining Displacement

Displacement depends only on the initial and final

positions.

If you drive from your house to the grocery store

and then to your friend’s house, the total distance

you have traveled is 10.7 miles; but your final

displacement from your initial position is -2.1 miles.


Defining Distance

The distance is the total length of travel. It is

always positive.

The distance from your house to the grocery store

and back is 8.6 miles. What is your final

displacement?


Scalar vs. Vector

Scalar: a numerical value (magnitude).

Vector: a quantity with both magnitude and

a direction in one or more dimensions.

Graphically, vectors are represented by

arrows.

• Distance is a scalar. Distance is always

positive.

• Position and Displacement are vectors.

Negative values indicate “in a negative

direction relative to the origin”.


Position vs. Displacement

• Position is relative to the origin (x=0).

• Displacement is relative your initial

position (xi): ∆x = xf – xi

Notation: ∆ = “delta” = “change in”

• Position and Displacement can be positive

or negative.


Question 2.1

A space shuttle takes off from the Kennedy

Space Center in Florida, circles Earth seven

times, and lands at Edwards Air Force Base

in California. A photographer travels from

Kennedy to Edwards to take a picture of the

astronauts.

Q: Who has the greater final displacement,

the astronauts, the photographer, or neither?

A: Neither, their displacement is the same.


Question 2.2 Displacement

Does the displacement of an object

depend on the specific location of

the origin of the coordinate system?

a) yes

b) no

c) it depends on the

coordinate system


Question 2.2 Displacement

10 20 30 40 50

30 40 50 60 70

40 -10 30x

60 30 30x

Does the displacement of an object

depend on the specific location of

the origin of the coordinate system?

a) yes

b) no

c) it depends on the

coordinate system

Because the displacement is

the difference between two

coordinates, the origin does

not matter.


Does the odometer in a car

measure distance or

displacement?

a) distance

b) displacement

c) both

Question 2.4 Odometer


Does the odometer in a car

measure distance or

displacement?

a) distance

b) displacement

c) both

If you go on a long trip and then return home, your odometer does not

measure zero, but it records the total miles that you traveled. That

means the odometer records distance.

Question 2.4 Odometer

Follow-up: how would you measure displacement in your car?


What is your position, displacement and

distance travelled at:

1. Initial time t = 0?

2. Time t = 2 sec?

3. Time t = 4 sec?


Speed and Velocity


Average Speed


Average Velocity


Example Problem 1

A kingfisher is a bird that catches fish by

plunging into the water from a height of

several meters. If a kingfisher dives from a

height of 8.0 m and reaches the water in 2.0

s, what was its speed?

Ave. speed = distance / elapsed time

= 8.0 m / 2.0 s

= 4.0 m/s


Example Problem 1b

If the kingfisher dives from a height of 7.0 m

with an average speed of 4.00 m/s, how

long does it take for it to reach the water?


4.00 m/s = 7.0 m / ∆t

∆t = 7.0 m / (4.00 m/s) = 1.75 s = 1.8 s


Example Problem 2

A car travels at 55 mph due north. On a

different road, a van travels at 55 mph due

east. Are their velocities the same?

No.

Their speeds are the same.

The magnitudes of their velocities are the

same.

Their directions are different.

Their velocities are different.


Example Problem 3: Speed

You drive you car at 30 mph for 1 hour and

then at 50 mph for another hour. What is

your average speed?


= 80 mi / 2 hr = 40 mph


Example Problem 3b: Speed You drive you car at 30.0 mph for 4.00 miles and then at

50.0 mph for another 4.00 miles. What is your average

speed?

Average speed = distance / elapsed time

Total distance is 8.00 mi. What is elapsed time?

t1 = 4.00 mi / (30.0 mi/h) = 0.133 h

t2 = 4.00 mi / (50.0 mi/h) = 0.0800 h

Ave. speed = 8.00 mi / (0.133 h + 0.080 h) = 37.6 mph

Average speed = “time-averaged” speed


Example Problem 4: Velocity

You start at position x = 10 m, and walk to

position x = 5 m in 10 seconds. What was

your average velocity?

Average velocity = (xf – xi) / (tf – ti)

= (5 m – 10 m) / 10 s

= -0.5 m/s



An athlete sprints 50.0 m in 6.00 s, and then walks

back to the starting line in 40.0 s. What is the

average velocity of (a) the sprint, (b) the walk, and

(c) the complete round-trip?

6.00 s

40.0 s



6.00 s

40.0 s


Example Problem 5b: Velocity

6.00 s

40.0 s

What is the athlete’s average speed?

Speed = d/Δt = 100.0 m / 46.0 s

= 2.17 m/s


Instantaneous Velocity and Speed

The instantaneous velocity is the average velocity

in the limit as the time interval becomes

infinitesimally short.

The instantaneous speed is the magnitude

(absolute value) of the instantaneous velocity.


If the position of a car is

zero, does its speed have to

be zero?

a) yes

b) no

c) it depends on

the position

Question 2.3 Position and Speed


If the position of a car is

zero, does its speed have to

be zero?

a) yes

b) no

c) it depends on

the position

No, the speed does not depend on position; it depends on the change

of position. Because we know that the displacement does not depend

on the origin of the coordinate system, an object can easily start at x =

–3 and be moving by the time it gets to x = 0.

Question 2.3 Position and Speed


Does the speedometer in a

car measure velocity or

speed?

a) velocity

b) speed

c) both

d) neither

Question 2.5 Speedometer


Does the speedometer in a

car measure velocity or

speed?

a) velocity

b) speed

c) both

d) neither

The speedometer clearly measures speed, not velocity. Velocity is a

vector (depends on direction), but the speedometer does not care

what direction you are traveling. It only measures the magnitude of

the velocity, which is the speed.

Question 2.5 Speedometer

Follow-up: how would you measure velocity in your car?


Question 2.7 Velocity in One Dimension

If the average velocity is non-zero over

some time interval, does this mean that

the instantaneous velocity is never zero

during the same interval?

a) yes

b) no

c) it depends


Question 2.7 Velocity in One Dimension

No!!! For example, your average velocity for a trip home

might be 60 mph, but if you stopped for lunch on the way

home, there was an interval when your instantaneous

velocity was zero, in fact!

If the average velocity is non-zero over

some time interval, does this mean that

the instantaneous velocity is never zero

during the same interval?

a) yes

b) no

c) it depends


Position-Time Graphs


Position-time graphs

• Plotting position data on the y-axis and

time data creates a position-time graph.

t, Time (s)

x, P

ositio

n (

m) t x

0 0

5 4

10 8



• The position-time graph can be used to

interpolate or extrapolate position or time.

t, Time (s)

x, P

ositio

n (

m) t x

0 0

5 4

10 8


On a position-time

graph, the slope of

the line indicates

the velocity,

v = (xf – xi) / (tf – ti)

= ∆x/ ∆t.



• On a position-time graph, the slope of a

line indicates the velocity, ∆x/∆t.

t, Time (s)

x, P

ositio

n (

m)

A straight

line

indicates

constant

velocity.

Zero

slope

indicates

zero

velocity.


Position-time graphs • The instantaneous velocity is the average

velocity ∆x/∆t in the limit as the time interval ∆t

becomes infinitesimally short.

t, Time (s)

x, P

ositio

n (

m)

The

magnitude

of the slope

indicates

the speed.

Note the

units of

the slope

are m/s.


The same motion, plotted one-dimensionally

and as an x-t graph:


Position-time graph: Example • Compute the velocity:

t, Time (s)

x, P

ositio

n (

m)


Position-time graph: Example 2 • When xf = xi, the mean velocity is zero:

t, Time (s)

x, P

ositio

n (

m)


Equation of Motion

for Constant Velocity


Equation of Motion

for Constant Velocity

Position can be described as a function of

time. For a constant or average velocity:

v = (xf – xi) / (tf – ti)

rearranges to

xf = xi + v × (tf – ti)

final position = initial position + ave. velocity × elapsed time


Equation of Motion 2 If velocity is constant, knowing the velocity and

position of an object at one point in time will predict

its position at any point in time.

x, P

ositio

n (

m)

t, Time (s)


Example

A skateboarder with an initial position of 1.5 m

moves with a constant velocity of 3.0 m/s. (a) Write

the position-time equation of motion for the

skateboarder. (b) What is the position of the

skateboarder at t = 2.5 s?

(a) xf = xi + v × (tf – ti)

xf = 1.5 m + (3.0 m/s) × t

(b) xf = 1.5 m + (3.0 m/s) × 2.5 s = 9.0 m.


Equation of Motion

The equation of motion for constant velocity

is a straight line (i.e. linear):

xf = xi + v × (tf – ti)

If we choose the origin such that ti = 0:

xf = xi + v × tf

x = xi + v t

y = b + m x


Intersecting Lines • When lines on a position-time graph intersect,

the two objects have the same location at the

same time.

t, Time (s)

x, P

ositio

n (

m)

An air

traffic

controller

needs to

look out

for this!


Example 2 You and a friend are at the airport, hurrying to the gate.

Your friend is 7.8 m ahead of you and walking at a speed of

2.3 m/s. You step on a walkway that moves you at 4.2 m/s.

How soon do you catch up to your friend? What is your

position then?

You: xi = 0 m, v = 4.2 m/s at ti = 0

Your friend: xi = 7.8 m, v = 2.3 m/s at ti = 0

x = 0 + 4.2 t

x = 7.8 + 2.3 t

Solve by substitution: 4.2 t = 7.8 + 2.3 t

1.9 t = 7.8 → t = 7.8/1.9 = 4.105263158 s = 4.1 s

x = 4.2 (4.105 s) = 17.241 m = 17 m


Question 2.13a Graphing Velocity I

t

x

The graph of position versus

time for a car is given below.

What can you say about the

velocity of the car over time?

a) it speeds up all the time

b) it slows down all the time

c) it moves at constant velocity

d) sometimes it speeds up and

sometimes it slows down

e) not really sure


Question 2.13a Graphing Velocity I

t

x

The graph of position versus




The car moves at a constant velocity

because the x vs. t plot shows a straight

line. The slope of a straight line is

constant. Remember that the slope of x

vs. t is the velocity!






e) not really sure


t

x






e) not really sure

The graph of position vs.




Question 2.13b Graphing Velocity II







e) not really sure

x

The graph of position vs.




The car slows down all the time

because the slope of the x vs. t graph is

diminishing as time goes on.

Remember that the slope of x vs. t is

the velocity! At large t, the value of the

position x does not change, indicating

that the car must be at rest. t

Question 2.13b Graphing Velocity II


Does Time Exist?

• Opinion 1: “The past no longer exists, the future does not

yet exist; only the present exists.”

– “Time” is how we measure change.

• Opinion 2: Time exists.

– How else could the speed of light (in m/s) be a

universal constant?

– All physical matter exists in 3 dimensions of space,

time and mass/energy.

• In any case:

– Time is a unique dimension; it is unidirectional and

moving.

– Practically, treating Time as a dimension is expedient:

it allows us to predict the future and diagnose the

past, which is what we want to know.


Car Lab

or Acceleration Lab p 48

chapter 2. one-dimensional kinematics -...

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