Representing MotionChapter

2

Representing Motion

Represent motion through the use of words, motion diagrams, and graphs.

Use the terms position, distance, displacement, and time interval in a scientific manner to describe motion.

Chapter

2

In this chapter you will:

Picturing Motion

Motion relates to change in place and time. You must be able to answer the questions of where and when an object is positioned to describe its motion.

In the figure below, the car has moved from point A to point B in a specific time period.

Movement Along a Straight Line

Section

2.1

Picturing MotionSection

2.1

Motion Diagrams

Click image to view movie.

Definitions

• Motion Diagram: a series of images showing the positions of a moving object at equal time intervals.

• Particle Model: a motion diagram in which the images are replaced with a series of single points.

Section 2.1 Self-CheckClick here: Section 2.1 Review

Where and When?

Define coordinate systems for motion problems.

Recognize that the chosen coordinate system affects the sign of objects’ positions.

Define displacement.

Determine a time interval.

Use a motion diagram to answer questions about an object’s position or displacement.

In this section you will:

Section

2.2

Where and When?

In the example of the runner, the origin, represented by the zero end of the measuring tape, could be placed 5 m to the left of the tree.

The motion is in a straight line, thus, your measuring tape should lie along that straight line. The straight line is an axis of the coordinate system.

Coordinate Systems

Section

2.2

Where and When?

You can indicate how far away an object is from the origin at a particular time on the simplified motion diagram by drawing an arrow from the origin to the point representing the object, as shown in the figure. The arrow is a position VECTOR.

Coordinate Systems

Section

2.2

Where and When?

Quantities that have both size, also called magnitude, and direction, are called vectors, and can be represented by arrows.

Vectors and Scalars

Section

2.2

Quantities that are just numbers without any direction, such as distance, time, or temperature, are called scalars.

To add vectors graphically, the length of a vector should be proportional to the magnitude of the quantity being represented. So it is important to decide on the scale of your drawings.

Where and When?

The vector that represents the sum of the other two vectors is called the resultant.

Vectors and Scalars

Section

2.2

The resultant always points from the tail of the first vector to the tip of the last vector. “Add vectors head-to-tail.”

Vector Addition

• Click here for vector practice:

• Vector Practice

Where and When?

The difference between the initial and the final times is called the time interval.

Time Intervals and Displacement

Section

2.2

The common symbol for a time interval is ∆t, where the Greek letter delta, ∆, is used to represent a change in a quantity.

Where and When?

The time interval is defined mathematically as follows:

Time Intervals and Displacement

Section

2.2

it = t tf

Although i and f are used to represent the initial and final times, they can be initial and final times of any time interval you choose.

Also of importance is how the position changes. The symbol d may be used to represent position.

In physics, a position is a vector with its tail at the origin of a coordinate system and its tip at the place where the object is located at that time.

Where and When?

The figure below shows ∆d, an arrow drawn from the runner’s position at the tree to his position at the lamppost.

Time Intervals and Displacement

Section

2.2

The change in position during the time interval between ti and tf

is called displacement.

Where and When?

The length of the arrow represents the distance the runner moved, while the direction the arrow points indicates the direction of the displacement.

Displacement is mathematically defined as follows:

Time Intervals and Displacement

Section

2.2

= f id d d

Displacement is equal to the final position minus the initial position.

Section Check

Differentiate between scalar and vector quantities?

Question 1

Section

2.2

Section Check

Answer 1

Section

2.2

Quantities that have both magnitude and direction are called vectors, and can be represented by arrows. Quantities that are just numbers without any direction, such as time, are called scalars.

Section Check

What is displacement?

Question 2

Section

2.2

A. The vector drawn from the initial position to the final position of the motion in a coordinate system.

B. The length of the distance between the initial position and the final position of the motion in a coordinate system.

C. The amount by which the object is displaced from the initial position.

D. The amount by which the object moved from the initial position.

Section Check

Answer: A

Answer 2

Section

2.2

Reason: Options B, C, and D are all defining the distance of the motion and not the displacement. Displacement is a vector drawn from the starting position to the final position.

Refer the figure below and calculate the time taken by the car to travel from one signal to another signal?

Question 3

Section

2.2 Section Check

A. 20 min

B. 45 min

C. 25 min

D. 5 min

Section Check

Answer: C

Answer 3

Section

2.2

Reason: Time interval t = tf - ti

Here tf = 01:45 and ti = 01:20

Therefore, t = 25 min

Position-Time Graphs

Develop position-time graphs for moving objects.

Use a position-time graph to interpret an object’s position or displacement.

Make motion diagrams, pictorial representations, and position-time graphs that are equivalent representations describing an object’s motion.

In this section you will:

Section

2.3

Position-Time Graphs

Position Time Graphs

Section

2.3

Click image to view movie.

Graphs of an object’s position and time contain useful information about an object’s position at various times and can be helpful in determining the displacement of an object during various time intervals.

Position-Time Graphs

Using a Graph to Find Out Where and When

Section

2.3

The data in the table can be presented by plotting the time data on a horizontal axis and the position data on a vertical axis, which is called a position-time graph.

To draw the graph, plot the object’s recorded positions. Then, draw a line that best fits the recorded points. This line represents the most likely positions of the runner at the times between the recorded data points.

Position-Time Graphs

Using a Graph to Find Out Where and When

Section

2.3

The symbol d represents the instantaneous position of the object—the position at a particular instant.

Position-Time Graphs

Considering the Motion of Multiple Objects

In the graph, when and where does runner B pass runner A?

Section

2.3

Step 1: Analyze the Problem

Position-Time GraphsSection

2.3

Considering the Motion of Multiple Objects

A position-time graph of an athlete winning the 100-m run is shown. Estimate the time taken by the athlete to reach 65 m.

Question 1

Section

2.3 Section Check

A. 6.0 s

B. 6.5 s

C. 5.5 s

D. 7.0 s

Section Check

Answer: B

Answer 1

Section

2.3

Reason: Draw a horizontal line from the position of 65 m to the line of best fit. Draw a vertical line to touch the time axis from the point of intersection of the horizontal line and line of best fit. Note the time where the vertical line crosses the time axis. This is the estimated time taken by the athlete to reach 65 m.

A position-time graph of an athlete winning the 100-m run is shown. What was the instantaneous position of the athlete at 2.5 s?

Question 2

Section

2.3 Section Check

A. 15 m

B. 20 m

C. 25 m

D. 30 m

Section Check

Answer: C

Answer 2

Section

2.3

Reason: Draw a vertical line from the position of 2.5 m to the line of best fit. Draw a horizontal line to touch the position axis from the point of intersection of the vertical line and line of best fit. Note the position where the horizontal line crosses the position axis. This is the instantaneous position of the athlete at 2.5 s.

From the following position-time graph of two brothers running a 100-m run, analyze at what time do both brothers have the same position. The smaller brother started the race from the 20-m mark.

Question 3

Section

2.3 Section Check

Section Check

Answer 3

Section

2.3

The two brothers meet at 6 s. In the figure, we find the intersection of line representing the motion of one brother with the line representing the motion of other brother. These lines intersect at 6 s and at 60 m.

How Fast?

Define velocity.

Differentiate between speed and velocity.

Create pictorial, physical, and mathematical models of motion problems.

In this section you will:

Section

2.4

The displacement, ∆d, is greater for the jogger in red for the same time interval, ∆t.

The slope of the position vs. time graph is steeper for the jogger in red.

How Fast?

Velocity

Section

2.4

How Fast?

The slopes of the two lines are found as follows:

Average Velocity

Section

2.4

d dt t

f i

f i

Red slope =

6.0 m 2.0 m3.0 s 1.0 s

=

= 2.0 m/s

d dt t

f i

f i

Blue slope =

3.0 m 2.0 m3.0 s 2.0 s

=

= 1.0 m/s

t t t

f i

f i

Δ =Δd d d

v Average Velocity

We get the formula for average velocity from these slope calculations:

How Fast?

What conclusion about velocity can you draw from this graph?

Average Velocity

Section

2.4

How Fast?

Instantaneous velocity: Velocity at a given instant.

The instantaneous velocity is the slope of a line tangent to the position vs. time graph at that particular instant.

Section

2.4

Instantaneous Velocity

How Fast?

• Equation of Motion for Average Velocity

Using Equations

Section

2.4

td v + di=

The position-time graph of a car moving on a street is as given here. What is the average velocity of the car?

Question 3

Section

2.4 Section Check

A. 2.5 m/s

B. 5 m/s

C. 8 m/s

D. 10 m/s

Section Check

Answer: A

Answer 3

Section

2.4

Reason: Average velocity of an object is the slope of the position-time graph.

End of Chapter

Representing MotionChapter

2