Chapter 2: 1-D Kinematics Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

Editing by Mr. Gehman

© 2007

The Cheetah: A cat that is built for speed. Its strength and agility allow it to sustain a top speed of over 100 km/h. Such speeds can only be maintained for about ten seconds.

Photo © Vol. 44 Photo Disk/Getty

Objectives: After completing this module, you should be able to:

• Define and apply concepts of average and instantaneous velocity and acceleration.

• Solve problems involving initial and final velocity, acceleration, displacement, and time.

• Demonstrate your understanding of directions and signs for velocity, displacement, and acceleration.

• Solve problems involving a free-falling body in a gravitational field.

Uniform Acceleration in One Dimension:

• Motion is along a straight line (horizontal, vertical or slanted).

• Changes in motion result from a CONSTANT force producing uniform acceleration.

• The cause of motion will be discussed later. Here we only treat the changes.

• The moving object is treated as though it were a point particle.

Reference Frames and Displacement

Any 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.

Reference Frames and 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.

Distance and Displacement

Distance is the length of the actual path taken by an object. Consider travel from point A to point B in diagram below:

Distance is a scalar quantity (no direction):

Contains magnitude only and consists of a number and a unit.

(70 m, 40 mi/h, 10 gal)

Distance and Displacement

Displacement is the straight-line separation of two points in a specified direction.

A vector quantity:

Contains magnitude AND direction, a number, unit & angle.

(40 m east; 8 km/h, N)

Reference Frames and Displacement

The displacement is written:

Left:

Displacement is positive.

Right:

Displacement is negative.

d d2 d1

Distance and Displacement

• For motion along x or y axis, the displacement is determined by the x or y coordinate of its final position. Example: Consider a car that travels 8 m, E then 12 m, W.

Net displacement D is from the origin to the final position:

What is the distance traveled? 20 m !!

12 m,W

D

D = 4 m, W

x 8 m,E

x = +8 x = -4

The Signs of Displacement

• Displacement is positive (+) or negative (-) based on LOCATION.

2 m

-1 m

-2 m

The displacement is the y-coordinate. Whether motion is up or down, + or - is based on LOCATION.

Examples:

The direction of motion does not matter!

Definition of Speed

• Speed is the distance traveled per unit of time (a scalar quantity).

v = = s

t

20 m

4 s

v = 5 m/s

Not direction dependent!

A B s = 20 m

Time t = 4 s

Examples of Speed

Light = 3 x 108 m/s

Orbit

2 x 104 m/s

Jets = 300 m/s Car = 25 m/s

Speed Examples (Cont.)

Runner = 10 m/s

Snail = 0.001 m/s

Glacier = 1 x 10-5 m/s

Definition of Velocity

• Velocity is the displacement per unit of time. (A vector quantity.)

v = 3 m/s, East

Direction required!

A B d = 12 m

Time t = 4 s

12 m

4 s

Dv

t

Average Speed & Velocity

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

Velocity includes directional information:

t

d

t

sv

t

d

t

Dv

Avg. Speed vs. Avg. Velocity

v = 10 m/s, East

Direction required!

s

m

t

sv

4

100

v = 25 m/s

Not direction dependent!

40

4

s mv

t s

Speed Example Exercise

s

m

s

m

s

m

t

dv 488.948766.9

54.10

00.100

hr

km

hr

s

m

km

s

mv 16.34

1

3600

1000

1488.9

Converting to km/hr:

FloJo, ’88 Olympics

Example 1. A runner runs 200 m, east, then changes direction and runs 300 m, west. If the entire trip takes 60 s, what is the average speed and what is the average velocity?

Recall that average speed is a function only of total distance and total time:

Total distance: s = 200 m + 300 m = 500 m

500 m

60 s

total pathAverage speed

time Avg. speed

8.33 m/s

Direction does not matter!

start

s1 = 200 m s2 = 300 m

Example 1 (Cont.) Now we find the average velocity, which is the net displacement divided by time. In this case, the direction matters.

xo = 0

t = 60 s x1= +200 m xf = -100 m 0fx x

vt

x0 = 0 m; xf = -100 m

100 m 01.67 m/s

60 sv

Direction of final displacement is to the left as shown.

Average velocity: 1.67 m/s, Westv

Note: Average velocity is directed to the west.

Example 2. A sky diver jumps and falls for 600 m in 14 s. After chute opens, he falls another 400 m in 150 s. What is average speed for entire fall?

600 m

400 m

14 s

150 s

A

B

600 m + 400 m

14 s + 150 s

A B

A B

x xv

t t

1000 m

164 sv 6.10 m/sv

Average speed is a function only of total distance traveled and the total time required.

Total distance/ total time:

Average Speed and Instantaneous Velocity

The instantaneous velocity is how fast and in what direction an object is moving in a particular instant. (v at point C)

The average speed depends ONLY on the distance traveled and the time required.

A

B s = 20 m

Time t = 4 s

C

The Signs of Velocity

First choose + direction; then v is positive if motion is with that direction, and negative if it is against that direction.

Velocity is positive (+) or negative (-) based on direction of motion.

- +

- +

+

Velocity in Position-Time Graphs

This is a graph of p vs. t for an object moving with const. velocity. If we take the slope of this line (“rise over run”) we get

Notice that the slope of the p-t curve can be (+) or (-).

velocityt

dslope