Physics 31

Lecture 5:Motion in One Dimension

With Changing Velocity

v(t) from x(t)

•

( )

in the time interval t

x

xv t average velocity

t

v(t) is slope of x(t)

( )( ) instantaneous velocity

dx tv t

dt

x(t) from v(t)

xx v t v(t)

ttend

What does x(t) look like?

v(t) = constant=10m/s

1. How far would the object travel in one second?2. How far in two seconds?

Slope is v = constant

t

x(t)

Acceleration

Acceleration is:

• The rate of change of velocity

• The slope of a velocity-versus-time graph

Slide 2-20

These four motion diagrams show the motion of a particle along the x-axis. Rank these motion diagrams by the magnitude of the acceleration. There may be ties.

Checking Understanding

Slide 2-21

These four motion diagrams show the motion of a particle along the x-axis. Which motion diagrams correspond to a positive acceleration? Which motion diagrams correspond to a negative acceleration?

Slide 2-22

Checking Understanding

These six motion diagrams show the motion of a particle along the x-axis. Rank the accelerations corresponding to these motion diagrams, from most positive to most negative. There may be ties.

Checking Understanding

Slide 2-23

Acceleration—Rate of Change of Velocity

( ) average rate of change of velocity

in the time t = average acceleration

( )( ) = slope of v(t) versus t =

instantaneous acceleration

xx

x

va t

t

dv ta t

dt

v(t) from a(t)

( ) constant

( ) ( ) ( )

( )

f

i

t

i i f it

i

a t a

v t v t a dt v t at at

v t a t

0( )x x xv t v a t

x(t) from v(t)

x(t) is the area under the v(t) vs t curveFor constant a

2

( )

area under curve is:

( ) ( ( ))

1( )

2

f

i

f

i

x ix x

x

t

i xt

t

i ix x i ix f x ft

v t v a t

v

x t x v t dt

x v a t dt x v t a t

In general, for constant acceleration

21( )

2i ox xx t x v t a t

Here is a motion diagram of a car moving along a straight stretch of road:

Which of the following velocity-versus-time graphs matches this motion diagram?

Quiz, question 1

A.

Slide 2-13

B. C. D.

Question 2

Slide 2-15

A graph of position versus time for a basketball player moving down the court appears like so:

Which of the following velocity graphs matches the above position graph?

A. B. C. D.

Question 3

• A car is moving with a constant velocity in the positive x-direction along a straight road. The magnitude of the velocity is v=12 m/s. Ten seconds later, it crashes into a wall and stops very suddenly.

• A) sketch v(t) versus t for the first 15 seconds.• B) sketch x(t) versus t for the first 15 seconds.

Homework due Monday

Reading 2: 3-4 Motion with constant accelerationProblems: 2: CQ1, CQ6, MC18, 1, 3, 5, 8, 12