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A L e a r n i n g

R e s o u r c e

f o r

S t u d e n t s

Don Metz, Ph.D.

I

N

MOTION

I

N

MOTION

A L e a r n i n g

R e s o u r c e

f o r

S t u d e n t s

Don Metz, Ph.D.

I

N

MOTION

I

N

MOTION

T a b l e o f

• Chapter C o n t e n t s
• Introduction
• • Introduction
• • Car Crash – Who is to Blame?
• Chapter
• Analyzing Motion
• • Position and Displacement
• • Instants and Intervals of Time
• • Uniform Motion
• • Instantaneous Velocity
• • Accelerated Motion
• • Real-life Motion
• Chapter
• Inertia
• • Natural Motion – Aristotle
• • Natural Motion – Galileo
  • the “Second Collision” • Newton’s First Law and
  • on an Inclined Plane • The Velocity of a Car
  • Unrestrained Occupant • Inertia and the
• Chapter
• Forces and Motion
• • Force and Acceleration
• • Mass and Acceleration
• • Force and Mass
• • Force and Direction
• • Action-Reaction Forces
• • The “Rocket” Car Race - Chapter - Momentum and Energy - • Momentum - • Impulse and Momentum - • Cushioning Devices - • Protecting Occupants - in a Collision • Momentum and Energy - Chapter - Braking - • Braking Distance - • Total Stopping Distance - • Reaction Time - Chapter - Driving Responsibly - • Case Study #1 - • Case Study #2

I N M O T I O N : A L e a r n i n g R e s o u r c e f o r S t u d e n t s 4

Introduction

At times, life can be a blur, everything moves by quickly, and people and machines are constantly on the go. Our understanding of motion is important in our everyday lives, especially as our modes of transportation become increasingly more sophisticated.

C L A S S A C T I V I T Y :

Tapping into Prior Knowledge What do YOU know about motion? Use a rotational graffiti activity and the following questions to express your ideas about motion.

The class is divided into groups of three

and each group is given a large piece of

newsprint (24" x 36" works nicely and

is available in pads from office supply

stores). Write each of the questions

above on the top of separate pieces of

newsprint. Each question is repeated on

another newsprint with the word "DRAW"

added. Using coloured markers, each

group has one minute to put their ideas

onto the page. At the end of every minute

the teacher calls out "rotate" and the

students pass their newsprint onto the next

group. When the groups have addressed

each question twice (one written response

and one drawing response), they post their

results on the wall for discussion.

Think About IT!

1. What does it mean to move? How does an object move? Give examples.
2. What skills and abilities do you need to drive a car?
3. What happens in a car crash?
4. Is stunt car driving or NASCAR driving dangerous?

Discuss your ideas about motion with your group and with your class. Have you ever been in a car collision? Do you know anyone who has been in a car collision?

Rotational Graffiti

FIGURE 1

FIGURE 2

Accident Report – Driver of Car A “My car was travelling east at a constant velocity. As I approached the intersection the light turned green so I kept going through the intersection at the same speed. Then, I heard a loud bang and my car spun to the left carrying it into the pedestrian walkway on the perpendicular street past the skateboarder. My car ended up jumping the curb on the far side of the corner. I wasn’t injured but everything took place so quickly I’m not entirely sure what happened.”

Accident Report – Driver of Car B

“I was driving south on Main Street when I wanted to check the name of the next street. So I signalled and moved from the median lane to the curb lane. I don’t know whether the signal lights were red or green or yellow as I was trying to read the street sign.”

Accident Report – Skateboarder “I was boarding east along the pedestrian walkway with my walkman on so I didn’t hear any sounds. I wasn’t paying much attention to the traffic when suddenly Car A spun into the walkway. I bailed and my board collided with the rear portion of Car A. I wasn’t hurt but I don’t listen to the walkman anymore when I’m riding.”

Accident Report – Motorcyclist “I was travelling south and had the green light. As Car B took the curb lane I signalled for a left turn then proceeded cautiously into the intersection to complete my turn. Car A ran a red light and was making a left turn when I collided with the front end of her car. I was thrown over the hood of the car and landed in the street. My helmet was not securely tied and it flew off my

head on impact. I suffered a concussion and remained in the hospital for several days.”

Police Report

Front end damage was extensive to the motorcycle. Car A had damage on the front and back fenders. The driver of Car A claimed that the damage on the front end of the car was from a previous fender bender. The motorcyclist claimed that he made the damage on the front end of Car A and that the skateboard damaged the rear of Car A. Oil drops, skid marks, and the motorcyclist’s helmet were found in the locations marked in Figure 4. The driver of Car A and the motorcyclist have conflicting stories concerning who was responsible for the accident. Therefore, we recommend that a “physics expert” be approached to investigate each driver’s claim.

I N M O T I O N : A L e a r n i n g R e s o u r c e f o r S t u d e n t s 6

Can you be a “physics expert”? As you study the principles of motion in the next few chapters, revisit your explanation of the car crash. Later in the course, we will return to examine the evidence again. Our understanding of motion will help determine exactly what happened in this real-life example.

FIGURE 5

Physics Experts Wanted

7 A n a l y z i n g M o t i o n

Analyzing Motion

P o s i t i o n a n d D i s p l a c e m e n t

We can easily describe distance by measuring from the origin with a ruler. Direction can be reported in many ways. It is common to use a coordinate line, that is, a line labelled -3, -2, -1, 0, +1, +2, +3 with the origin at 0, as shown in the diagram above. In this case, we use the plus sign (+) to indicate a position to the right of the origin and a minus sign (-) to indicate a position left of the origin. There are other ways to describe direction. For instance, we could use a compass or a direction finder (north, south, east, west).

In this course, we will make describing direction very easy by restricting our motion along a single straight line. In this way, you can describe motion using a coordinate line or by using common terms like right and left, forward and backward, or, if your line is vertical, up and down. Any quantity that is described using magnitude and direction is called a vector quantity. In this text, when vector quantities are represented in symbolic form, they will be bolded.

Let’s start here

What does it mean to move? First of all, if you stay in one place without changing your position, we would say you are standing still. You’re not moving! In order to move, you must change your position. So, get off the couch and move! If you want to describe the movement of an object (for example, a person running or a vehicle in motion), you must be able to describe its position. In order to communicate our information to other persons, everyone must agree on a reference point, called the origin, from which we begin to take measurements. Position is the distance and direction an object is located from an origin.

They call it inertia…

9

PRACTICE

1. The following data represent the initial (d 1 ) and final (d 2 ) positions of a car, bicycle, pedestrian, and skateboarder.

a) Draw a number line and label an origin as point “0”. Mark the initial position of each object above the line.

b) Mark the final position of each object below the number line.

c) Calculate the displacement of each object.

d) What is happening? If each displacement takes place in the same period of time, write a paragraph to describe the motion of each object.

2. The dispatcher of a courier service receives a message from Truck A that reports a position of +5 after a displacement of +2. What was the initial position of Truck A? First solve the problem using a number line, and then solve the problem using an equation. 3. Two taxis are travelling along Pembina Highway in opposite directions. Taxi A changes its position from +6 to +10 during the same time as Taxi B moves from +6 to +1. Draw a diagram to show the initial and final positions of each taxi. 4. Calculate the displacements of each taxi in question #3. 5. What can you conclude about the speed of the taxis? 6. How would the position of the taxis change if you decided to move your origin? How does the displacement of the taxis change if you decide to move your origin?

Analyzing Motion

Car Bicycle Pedestrian

d 1

d 2

Skateboarder

+2 m

+14 m

+7 m

+2 m

-1 m

+2 m

+4 m

-1 m

We know that cars move at different speeds; they speed up and slow down. In the previous taxi example, we know that Taxi B was moving faster because it travelled a greater distance in the same period of time as Taxi A. However, we still do not know the speed of the taxi. Was traffic moving slowly or quickly? An instant of time is a reading on a clock, such as 10:15 or 36.2 seconds. In order to know how fast an object is moving, we need to know the time it took for the car to move from one position to another (change in time). An interval of time is the difference between two such clock readings. Thus,

I N M O T I O N : A L e a r n i n g R e s o u r c e f o r S t u d e n t s 10

I n s t a n t s a n d I n t e r v a l s o f T i m e

Think About IT!

1. Describe several real-life examples of an interval of time.
2. Determine how many hours is an interval of 10 seconds? What made this calculation a difficult one for you?

Time is a quantity for which direction is not required. Quantities that describe magnitude only are called scalars.

In order to answer the question “How fast is the taxi going?”, we need to collect information about the position of the taxi at different points in time.

Interval of time = time 2 - time 1

∆t = t 2 - t 1

or in SYMBOLIC FORM

Reminder

“ ∆ ” (^) means

“Change in”

On most highways there is a posted speed limit. (Do you know what the speed limit is if no sign is posted?) Cars that travel down the highway at this speed are said to be moving in uniform motion. That is, their motion is constant. A picture of Car A travelling down a highway at a constant speed is shown in Diagram A.

Each frame represents a picture of the car at one-second intervals. Notice that the spaces are equal for equal time intervals. Measure the distance between each interval and complete Table A. Graph your results with the position on the y axis and the time on the x axis. Repeat the procedure using measurements from Diagram B and answer the questions.

I N M O T I O N : A L e a r n i n g R e s o u r c e f o r S t u d e n t s 12

U n i f o r m M o t i o n

Diagram A

Diagram B

Position (cm) Time (s)

Table A

Position (cm) Time (s)

Table B Think About IT!

1. What is the difference between Graph A and Graph B?
2. How do the spacings of the cars in the two diagrams compare to the plotted points on the graphs?

Sample Only. Do not write here.

Sample Only. Do not write here.

13

◗ S l o p e

The spacings between the dots in our frame-by- frame analysis are reflected in the steepness of the line on the graphs. The steeper the line, the larger the spaces. That is, Car B travelled a greater distance in 1 second than Car A. In other words, Car B was going faster than Car A. “How fast” a car moves is called the speed of the car and is displayed on the car’s speedometer.

Velocity is the term physicists use to describe how fast and in what direction an object moves. Velocity is defined as:

This definition is true for objects whose velocities are not changing. In real life, it is very difficult to find objects that move exactly uniformly, so we often assume that an object has a constant velocity even if it does vary somewhat. Consequently, we call this form the average velocity and in symbolic form we write:

Velocity is a vector and always has a direction. In your answers, you can use common terms like right, left, forward and backward, or if you use a coordinate line, + or - signs.

The steepness of the line on the graph is called the slope of the line. The slope of any straight line (including the roof on your house!) is always constant. Numerically, this constant is the ratio of the rise (the vertical displacement (∆y)) and the corresponding run (the horizontal displacement (∆x)). We can calculate slope by using the formula

(We will do this in detail on the next page). In our motion example, the position ( d ) is on the y axis and time (t) is on the x axis. Therefore, the slope is ∆ d /∆t. That is, the slope of the line is the velocity of the object.

A n a l y z i n g M o t i o n

Velocity =

Change in position

Change in time

v avg =

∆ d

∆t

Slope =

Rise

Run

or

Slope =

∆y

∆x

Think About IT! (^) 1. Make a concept map to link the following terms together:

Include some terms of your own.

• slope,
• velocity,
• speed,
• ∆ d ,
• ∆t,
• steepness,
  • constant,
  • displacement,
  • rise,
  • run.

15 A n a l y z i n g M o t i o n

Converting from m/s to km/h

Example: Convert 4.0 m/s to km/h

1 m = 0.001 km 4.0 m = 0.004 km

That is, divide metres by 1 000

1 h = 60 min and 1 min = 60 seconds

1 h = 60 x 60 = 3 600 seconds

1 second = hour

4.0 m/s = km/h

Conversion Shortcut:

To convert m/s to km/h just multiply m/s by 3.6!

3. A skateboarder is coasting at a velocity of 2 m/s away from the corner. If we let the corner be the origin, how far will the boarder travel in 3.5 seconds? 4. In terms of the displacement of a vehicle on a highway, what does speeding mean? 5. A mini-V car rolls off a ramp with a constant velocity of 1.5 m/s onto a horizontal track. The end of the ramp is at position -12 cm. If the car reaches the end of the track in 0.4 seconds find the length of the track. Include a diagram and label the origin.

1 s

4.0 m

AND

4.0 m/s = x km/h

4.0 m/s = x 3.6 km/h

4.0 m/s = 14.4 km/h

Average velocity describes the velocity of an object during an interval of time. Instantaneous velocity is the velocity at a specific time. For uniform motion, the instantaneous velocity is always the same as the average velocity, which is why we call it uniform. For uniform motion, the position-time graphs are straight lines. However, the position-time graphs for non-uniform motion are curves. Diagram A is the position-time graph of object in non-uniform motion. Can you tell from the graph when the object is going slow or going fast?

Recall that the velocity is the slope of the position-time graph. So, the object is going fast when the slope is very steep and is going more slowly when the slope is more gradual. (Diagram B)

Point A – steep slope, going fast

Point B – gradual slope, going slow

Point C – steep slope, going fast but in the opposite direction.

In most cases we can closely approximate an instantaneous velocity by choosing a small enough interval of time such that the graph is almost a straight line. For example, using the position-time graph, if we choose an interval of time from t = 0.2 h to t = 0.3 h and draw a line between these two points, this straight line closely approximates the curve. (Diagram C)

I N M O T I O N : A L e a r n i n g R e s o u r c e f o r S t u d e n t s 16

I n s t a n t a n e o u s V e l o c i t y

Position vs. Time

Position vs. Time

Position vs. Time

Diagram A

Diagram B

Diagram C

As you drive your car from a stop light or a parking spot, you must gradually increase your speed from zero to the posted speed limit. When a vehicle speeds up or slows down, we say that it is accelerating.

I N M O T I O N : A L e a r n i n g R e s o u r c e f o r S t u d e n t s 18

A c c e l e r a t e d M o t i o n

C L A S S A C T I V I T Y :

Remember the dots from the mini-V car as it rolled down the ramp? Did you notice that the spaces were increasing for equal time intervals? As the car moves faster, the distance between the dots increases. Measure the distance between the dots for each interval and complete Table C (you can use your results from page 11 or use the diagram above). Graph your results with the position on the y axis and the time on the x axis.

Think About IT!

1. Does your graph describe uniform or non-uniform motion?
2. What can you say about the spacing of the dots and the velocity of the car?
3. What can you say about the acceleration of the car?

It is very easy to mix up the terms velocity and acceleration.

Position (cm/s) Time (s)

Table C

Sample Only. Do not write here.

Remember

that velocity is a

change in position.

In addition, an object

must change its velocity

to accelerate.

19

From your position versus time graph, the line curves upwards because the spacing between the dots is increasing. We can draw reference lines on the graph to show how the points on the curve correspond to the spacing of the dots.

In order to investigate the changes in velocity we must graph the instantaneous velocities for several different times. First, complete Table D to find the average velocity for each interval of time.

A n a l y z i n g M o t i o n

Position

(cm/s)

Time

(s)

Table D

Average Velocity

(cm/s)

In order to find how velocity changes, we must graph instantaneous velocity versus time.

1. First, calculate the average

velocity for each interval to

complete Table D.

2. Remember that the average

velocity of an interval

closely approximates the

instantaneous velocity at

the midpoint of the interval.

Use Table D to complete

Table E and then graph

velocity versus time.

Graphing Instructions

Position vs. Time

Sample Only. Do not write here.