Chapter 2: Motion in One Dimension

Section 1: Displacement & Velocity

Mechanics

• Kinematics Describes the motion of objects without looking at the cause of the

motion• Dynamics

Deals with the effect that forces have on motion

• Motion the change of an object’s position relative

to some reference point

Describing MotionMost basic

information needed: position and time

Whether or not an object is in motion depends on the reference point you choose.

Measuring Motion

• Scalar quantity Have magnitude (numerical value) but no direction

• Vector Have magnitude and direction

“Quantity” is the root word for quantitative… you get a number answer

Distance

• Distance (d) – how far an object travels.– Does not depend on direction.

• Imagine an ant crawling along a ruler.

• What distance did the ant travel?– d = 3 cm

cm

0 1 2 3 4 5 6 7 8 9 10

Distance

• Distance does not depend on direction.• Here’s our intrepid ant explorer again.

• Now what distance did the ant travel?– d = 3 cm

• Does his direction change the answer?

cm

0 1 2 3 4 5 6 7 8 9 10

Distance

• Distance does not depend on direction.• Let’s follow the ant again.

• What distance did the ant walk this time?• d = 7 cm

cm

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• Objects that are “at rest” are not in motion.• Displacement – change in position of an

object – measures how far an object has moved from its starting location

Note: ∆ is the Greek symbol “delta”. It means “a change in”.

xf means final position. xi means initial position.

Displacement: ∆x = xf – xi

Examples of directions:+ and –N, S, E, WAngles

In order to define displacement, we need a direction

(depends on your frame of reference)

Displacement vs. Distance

• Example of distance:– The ant walked 3 cm.

• Example of displacement:– The ant walked 3 cm EAST.

An object’s distance traveled and its displacement aren’t

always the same!

cm

0 1 2 3 4 5 6 7 8 9 10

+-

Displacement• Let’s revisit our ant, and this time we’ll find his

displacement.

• Distance: 3 cm• Displacement: +3 cm

– The positive gives the ant a direction!

Displacement

• Find the ant’s displacement again.– Remember, displacement has direction!

• Distance: 3 cm• Displacement: -3 cm

cm

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+-

Displacement

• Find the distance and displacement of the ant.

• Distance: 7 cm• Displacement: +3 cm

cm

0 1 2 3 4 5 6 7 8 9 10

+-

Displacement vs. Distance

• An athlete runs around a track that is 100 meters long three times, then stops.– What is the athlete’s distance and displacement?

• Distance = 300 m• Displacement = 0

m• Why?

• Displacement can be either positive or negative…depending on your frame of reference.

• The right (or east) will be considered positive unless otherwise stated. Likewise with up (north).

• The left (or west) will be considered negative unless otherwise stated. Likewise with down (south).

PositiveNegative

∆x = 8 cm∆x = -4 cm

With your partner, discuss the following:1) What is this object’s position at 1 minute?2) What is the object’s position at 2 minutes?3) What is its displacement between 1 min and 5 min?4) Describe the motion of this object.

Speed vs Velocity

Speed

• Speed (s) – Rate at which an object is moving.• speed = distance / time• s = d/t• Like distance, speed does not depend on direction.

Speed

• A car drives 100 meters in 5 seconds.

• What is the car’s average speed?– s = d/t– s = (100 m) / (5 s) = 20 m/s

100 m

1 s2 s3 s4 s5 s

• Average velocity – the total displacement divided by the total amount of time during the displacement.

• Velocity may be + or - , depending on the displacement.

Velocity: v = ∆x = (xf-xi) ∆t (tf – ti)

Why refer to it as average velocity?Example:Suppose you traveled from your house to school…a

distance of 4.0 km. It took you .20 hours (12 minutes) because of heavy traffic.

Your avg velocity would be:v = 4.0 km / .20 hv = 20 km/hDid you travel at that exact speed for the entire trip?

Of course not.Because of this, velocity is an average.

Velocity and speed are not the same:Velocity requires direction.Speed refers to the numerical value of

velocity.

Avg. Speed and Avg. Velocity

• I walk 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average speed and average velocity.

Average speed: 0.50m/sDisplacement is 0 meters, average velocity is 0m/s

Pulling It All Together

• Back to our ant explorer!

– Distance traveled: 7 cm– Displacement: +3 cm– Average speed: (7 cm) / (5 s) = 1.4 cm/s– Average velocity: (+3 cm) / (5 s) = +0.6 cm/s

cm

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+-1 s2 s3 s4 s5 s

Example

• A crow flies east in a straight line. It begins flying at 3:04s, and comes to a rest at 3:11s. It was flying at an

average velocity of 1.5 m/s. How far did the crow move from its initial position (in meters)?

• Using the velocity equation, we can derive another equation to find an object’s position

(Final) Displacement: xf = v∆t + xi

With your partner…1) Use the base velocity equation to derive an equation to

solve for the final displacement of an object.

Velocity can be determined using a position vs time graph.

The slope of the line corresponds to the velocity.

Position vs Time

Object 1

Object 2

Object 3

Time

Pos

ition

How could we describe the motion of these objects?

ReviewConstant VelocityPositive Velocity

Positive VelocityChanging Velocity

(acceleration)

Slow, Rightward(+)Constant Velocity

Fast, Rightward(+)Constant Velocity

Graphing !x

t

A

B

C

A … Starts at home (origin) and goes forward slowly

B … Not moving (position remains constant as time progresses)

C … Turns around and goes in the other direction quickly, passing up home

1 – D Motion

What is the AVERAGE speed of the bass boat depicted in the

graph?

Bass Boat Speed

0

20

50

65 65 65

85

100

125

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10

Time (seconds)

Dis

tan

ce

(m

ete

rs)

Average speed is taking the total distance traveled (0 to 125 meters), and dividing by the total time (1 to 9 seconds) it takes.

Bass Boat Speed

0

20

50

65 65 65

85

100

125

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10

Time (seconds)

Dis

tan

ce

(m

ete

rs)

What is the instantaneous

speed of the bass boat at t=7 seconds?

Instantaneous Speed = 85 meters = 12.1 m/s 7 seconds

Example

• Sketch a position vs. time and a velocity vs. time graph for the following:

Your mother walks five meters to the kitchen at a velocity of 2 m/s, pauses by the refrigerator for three seconds, then walks back with a plate full of potato salad at a velocity of 1 m/s.

• At track practice, the coach makes the team members run back and forth between two lines three times

• A car is driving down the road at 55 mph, and after getting a flat tire moves at 35 mph