Chapter 2 One-Dimensional Kinematics - University of …nsmn1.uh.edu/hpeng5/Peng02a_LectureOutline.pdf · · 2013-01-182-1 Position, Distance, and Displacement Displacement is the

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Chapter 2

One-Dimensional Kinematics

Description of motion in one dimension


Units of Chapter 2• Position, Distance, and Displacement

• Average Speed and Velocity

• Instantaneous Velocity

• Acceleration

• Motion with Constant Acceleration

• Applications of the Equations of Motion

• Freely Falling Objects


2-1 Position, Distance, and Displacement

Before describing motion, you must set up a coordinate system – define an origin and a positive direction.

Position: described by x



The distance is the total length of travel; if you drive from your house to the grocery store and back, you have covered a distance of 8.6 mi.



Displacement is the change in position. If you drive from your house to the grocery store and then to your friend’s house, your displacement is 2.1 mi and the distance you have traveled is 10.7 mi.

displacement change in position final position initial positiondisplacement f i

= = −= = −Δ Δx x x

You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement?

a) yes

b) no

Question 2.1 Walking the Dog

You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement?

a) yes

b) no

Yes, you have the same displacement. Because you and your dog had the same initial position and the same final position, then you have (by definition) the same displacement.

Question 2.1 Walking the Dog

Follow-up: have you and your dog traveled the same distance?


2-2 Average Speed and Velocity

The average speed is defined as the distance traveled divided by the time the trip took:

Average speed = distance / elapsed timeIs the average speed of the red car 40.0 mi/h, more than 40.0 mi/h, or less than 40.0 mi/h?


2-2 Average Speed and VelocityThe average speed is defined as the distance traveled divided by the time the trip took:

Average speed = distance / elapsed timeIs the average speed of the red car 40.0 mi/h, more than 40.0 mi/h, or less than 40.0 mi/h?

hhmi

mit )0.30/00.4(/0.30

00.41 ==

hhmi

mit )0.50/00.4(/0.50

00.42 ==

htt 213.021 =+ Average speed = hmihmih

mi /40/6.37213.08

<=



Average velocity = displacement / elapsed time

If you return to your starting point, your average velocity is zero.

average velocitydisplacementelapsed time

avf i

f i

=

= =−−

v xt

x xt t

ΔΔ



Graphical Interpretation of Average Velocity

The same motion, plotted one-dimensionally and as an x-t graph:


2-3 Instantaneous Velocity

Definition:

(2-4)

This means that we evaluate the average velocity over a shorter and shorter period of time; as that time becomes infinitesimally small, we have the instantaneous velocity.


2-3 Instantaneous VelocityThis plot shows the average velocity being measured over shorter and shorter intervals. The instantaneous velocity is tangent to the curve.


avf i

f i

=

= =−−

v xt

x xt t

ΔΔ

v xtt

=→

limΔ

ΔΔ0


2-3 Instantaneous Velocity

Graphical Interpretation of Average and Instantaneous Velocity


2-4 Acceleration

Average acceleration:

(2-5)


2-4 AccelerationGraphical Interpretation of Average and Instantaneous Acceleration:

a vtt

=→

limΔ

ΔΔ0

a vt

v vt tav

f i

f i= =

−−

ΔΔ


2-4 Acceleration

Acceleration (increasing speed) and deceleration (decreasing speed) should not be confused with the directions of velocity and acceleration:


Units of Chapter 2• Position, Distance, and Displacement

• Average Speed and Velocity

• Instantaneous Velocity

• Acceleration

• Motion with Constant Acceleration

• Applications of the Equations of Motion

• Freely Falling Objects


Review: 2-3 Instantaneous VelocityThis plot shows the average velocity being measured over shorter and shorter intervals. The instantaneous velocity is tangent to the curve.


avf i

f i

=

= =−−

v xt

x xt t

ΔΔ

v xtt

=→

limΔ

ΔΔ0


Review: 2-4 AccelerationGraphical Interpretation of Average and Instantaneous Acceleration:

a vtt

=→

limΔ

ΔΔ0

a vt

v vt tav

f i

f i= =

−−

ΔΔ

Question 2.13aQuestion 2.13a Graphing Velocity IGraphing 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 timeb) it slows down all the timec) it moves at constant velocityd) sometimes it speeds up and

sometimes it slows downe) not really sure

Question 2.13aQuestion 2.13a Graphing Velocity IGraphing 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 isconstant. Remember that the slope of xvs. t is the velocity!



t

x



The graph of position vs.




Question 2.13b Graphing Velocity II



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

Consider the line labeled A in

the v vs. t plot. How does the

speed change with time for

line A?

a) decreases

b) increases

c) stays constant

d) increases, then decreases

e) decreases, then increases

Question 2.14a v versus t graphs I

v

t

A

B

Consider the line labeled A in



line A?

a) decreases

b) increases

c) stays constant



In case A, the initial velocity is positive and the magnitude of the velocity continues to increase with time.

Question 2.14a v versus t graphs I

v

t

A

B

Consider the line labeled B in



line B?

a) decreases

b) increases

c) stays constant



Question 2.14b v versus t graphs II

v

t

A

B

Consider the line labeled B in



line B?

a) decreases

b) increases

c) stays constant



In case B, the initial velocity is positive but the magnitude of the velocity decreases toward zero. After this, the magnitude increases again, but becomes negative, indicating that the object has changed direction.

Question 2.14b v versus t graphs II

v

t

A

B


2-5 Motion with Constant Acceleration

If the acceleration is constant, the velocity changes linearly:

(2-7)

Average velocity:

Note: valid only for constant acceleration



Average velocity:

(2-9)

Position as a function of time:

(2-10)

(2-11)

Velocity as a function of position:

(2-12)

x x v t= +0 av

v v at= +0



The relationship between position and time follows a characteristic curve.

Chapter 2 One-Dimensional Kinematics - University of …nsmn1.uh.edu/hpeng5/Peng02a_LectureOutline.pdf · · 2013-01-182-1 Position, Distance, and Displacement Displacement is the

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