chapter 2: representing motion physics principles and problems

Chapter 2: Representing MotionChapter 2: Representing Motion

PHYSICS Principles and

Problems

You can use displacement and velocity to describe an object’s motion.

BIG IDEA

CHAPTER

2 Representing Motion

Section 2.1 Picturing Motion

Section 2.2 Where and When?

Section 2.3 Position-Time Graphs

Section 2.4 How Fast?

CHAPTER

2 Table Of Contents

ExitClick a hyperlink to view the corresponding slides.

MAIN IDEA

You can use motion diagrams to show how an object’s position changes over time.

Essential Questions• How do motion diagrams represent motion?

• How can you use a particle model to represent a moving object?

SECTION2.1

Picturing Motion

Review Vocabulary

• Model a representation of an idea, event, structure or object to help people better understand it.

New Vocabulary• Motion diagram• Particle model

SECTION2.1

Picturing Motion

• Perceiving motion is instinctive—your eyes pay more attention to moving objects than to stationary ones. Movement is all around you.

• Movement travels in many directions, such as the straight-line path of a bowling ball in a lane’s gutter, the curved path of a tether ball, the spiral of a falling kite, and the swirls of water circling a drain.

All Kinds of Motion

SECTION2.1

Picturing Motion

• When an object is in motion, its position changes. Its position can change along the path of a straight line, a circle, an arc, or a back-and-forth vibration.

All Kinds of Motion (cont.)

SECTION2.1

Picturing Motion

• Straight-line motion follows a path directly between two points without turning left or right.

–Ex. Forward and backward, up and down, or north and south.

• A description of motion relates to place and time. You must be able to answer the questions of where and when an object is positioned to describe its motion.

SECTION2.1

Picturing Motion

All Kinds of Motion (cont.)

Motion Diagrams

Click image to view movie.

SECTION2.1

Picturing Motion

Explain how applying the particle model produces a simplified version of a motion diagram?

SECTION2.1

Section Check

Answer

Answer: Keeping track of the motion of the runner is easier if we disregard the movements of the arms and the legs, and instead concentrate on a single point at the center of the body. In effect, we can disregard the fact that the runner has some size and imagine that the runner is a very small object located precisely at that central point. A particle model is a simplified version of a motion diagram in which the object in motion is replaced by a series of single points.

SECTION2.1

Section Check

Which statement describes best the motion diagram of an object in motion?

A. a graph of the time data on a horizontal axis and the position on a vertical axis

B. a series of images showing the positions of a moving object at equal time intervals

C. a diagram in which the object in motion is replaced by a series of single points

D. a diagram that tells us the location of the zero point of the object in motion and the direction in which the object is moving

SECTION2.1

Section Check

Reason: A series of images showing the positions of a moving object at equal time intervals is called a motion diagram.

SECTION2.1

Answer

Section Check

What is the purpose of drawing a motion diagram or a particle model?

A. to calculate the speed of the object in motion

B. to calculate the distance covered by the object in a particular time

C. to check whether an object is in motion

D. to calculate the instantaneous velocity of the object in motion

SECTION2.1

Section Check

Reason: In a motion diagram or a particle model, we relate the motion of the object with the background, which indicates that relative to the background, only the object is in motion.

SECTION2.1

Answer

Section Check

MAIN IDEA

A coordinate system is helpful when you are describing motion.

Essential Questions• What is a coordinate system?

• How does the chosen coordinate system affect the sign of objects’ positions?

• How are time intervals measured?

• What is displacement?

• How are motion diagrams helpful in answering questions about an object’s position or displacement?

SECTION2.2

Where and When?

Review Vocabulary

• Dimension extension in a given direction;one dimension is along a straight line; three dimensions are height, width and length.

New Vocabulary

• Coordinate system

• Origin

• Position

• Distance

• Magnitude

• Vector

• Scalar

• Time interval

• Displacement

• Resultant

SECTION2.2

Where and When?

• A coordinate system tells you the location of the zero point of the variable you are studying and the direction in which the values of the variable increase.

• The origin is the point at which both variables have the value zero.

Coordinate Systems

SECTION2.2

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.

Coordinate Systems (cont.)

SECTION2.2

Where and When?

• 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.

SECTION2.2

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.

SECTION2.2

Where and When?


• The two arrows locate the runner’s position at two different times.

• Because the motion in the figure below is in one direction, the arrow lengths represent distance.

SECTION2.2

Where and When?


• The length of how far an object is from the origin indicates its distance from the origin.

SECTION2.2

Where and When?


• A position 9 m to the left of the tree, 5 m left of the origin, would be a negative position, as shown in the figure below.

SECTION2.2

Where and When?


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

– Vector quantities will be represented by boldface letters.

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

– Scalars quantities will be represented by regular letters.

Vectors and Scalars

SECTION2.2

Where and When?

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

Vectors and Scalars (cont.)

SECTION2.2

Where and When?

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

SECTION2.2

Where and When?


• The time interval is defined mathematically as follows:

• 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.

• The time interval is a scalar because it has no direction.

SECTION2.2

Where and When?


• The figure below shows the position of the runner at both the tree and the lamppost.

• These arrows have magnitude and direction.

• Position is a vector with the arrow’s tail at the origin and the arrow’s tip at the place.

SECTION2.2

Where and When?


• The symbol x is used to represent position vectors mathematically.

• Xi represents the position at the tree, xf represents the position at the lamppost and ∆x, represents the change in position, displacement, from the tree to the lamppost.

SECTION2.2

Where and When?


• Displacement is defined mathematically as:

• Remember that the initial and final positions are the start and end of any interval you choose, so a plus and minus sign might be used to indicate direction.

∆x = xf - xi

SECTION2.2

Where and When?


• A vector that represents the sum o f two other vectors is called a resultant.

• The figure to the right shows how to add and subtract vectors in one dimension.

SECTION2.2

Where and When?


• To completely describe an object’s displacement, you must indicate the distance it traveled and the direction it moved. Thus, displacement, a vector, is not identical to distance, a scalar; it is distance and direction.

• While the vectors drawn to represent each position change, the length and direction of the displacement vector does not.

• The displacement vector is always drawn with its flat end, or tail, at the earlier position, and its point, or tip, at the later position.

SECTION2.2

Where and When?


Differentiate between scalar and vector quantities.

SECTION2.2

Section Check

Reason: 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.

SECTION2.2

Answer

Section Check

What is displacement?

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

B. 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

SECTION2.2

Section Check

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.

SECTION2.2

Answer

Section Check

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

A. 20 min

B. 45 min

C. 25 min

D. 5 min

SECTION2.2

Section Check

Reason: Time interval t = tf – ti

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

Therefore, t = 25 min

SECTION2.2

Answer

Section Check

MAIN IDEA

You can use position-time graphs to determine an object’s position at a certain time.

Essential Questions• What information do position-time graphs provide?

• How can you use a position-time graph to interpret an object’s position or displacement?

• What are the purposes of equivalent representations of an object’s motion?

SECTION2.3

Position-Time Graphs

Review Vocabulary

• Intersection a point where lines meet and cross.

New Vocabulary

• Position-time graph

• Instantaneous position

SECTION2.3


Finding Positions

Click image to view movie.

SECTION2.3


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

Finding Positions (cont.)

SECTION2.3


• 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.

SECTION2.3



• 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.

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

SECTION2.3



• Words, pictorial representations, motion diagrams, data tables, and position-time graphs are all representations that are equivalent. They all contain the same information about an object’s motion.

• Depending on what you want to find out about an object’s motion, some of the representations will be more useful than others.

SECTION2.3



Multiple Objects on a Position-Time Graph

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

SECTION2.3


Step 1: Analyze the Problem

Multiple Objects on a Position-Time Graph (cont.)

Restate the questions.

Question 1: At what time do A and B have the same position?

Question 2: What is the position of runner A and runner B at this time?

SECTION2.3


Step 2: Solve for the Unknown

SECTION2.3



Question 1

In the figure, examine the graph to find the intersection of the line representing the motion of A with the line representing the motion of B.

These lines intersect at 45 s.

SECTION2.3



Question 2

In the figure, examine the graph to find the intersection of the line representing the motion of A with the line representing the motion of B.

The position of both runners is about 190m from the origin.

SECTION2.3



B passes A about 190 m beyond the origin, 45.0 s after A has passed the origin.

SECTION2.3



The steps covered were:

Step 1: Analyze the Problem

Restate the questions.


SECTION2.3


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.

A. 6.0 s

B. 6.5 s

C. 5.5 s

D. 7.0 s

SECTION2.3

Section Check

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.

SECTION2.3

Answer

Section Check

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?

A. 15 m

B. 20 m

C. 25 m

D. 30 m

SECTION2.3

Section Check

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.

SECTION2.3

Answer

Section Check

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

SECTION2.3

Section Check

Reason: The two brothers meet at 6 s. In the figure, we find the intersection of lines 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.

SECTION2.3

Answer

Section Check

MAIN IDEA

An object’s velocity is the rate of change in its position.

Essential Questions• What is velocity?

• What is the difference between speed and velocity?

• How can you determine an object’s average velocity from a position-time graph?

• How can you represent motion with pictorial, physical, and mathematical models?

SECTION2.4

How Fast?

Review Vocabulary

• Absolute value magnitude of a number, regardless of sign.

New Vocabulary

• Average velocity

• Average speed

• Instantaneous velocity

SECTION2.4

How Fast?

• Suppose you recorded two joggers in one motion diagram, as shown in the figure below. The position of the jogger wearing red changes more than the of the jogger wearing blue

Velocity and Speed

• For a fixed time, the magnitude of the displacement (∆x), is greater for the jogger in red.

• If each jogger travels 100m, the time interval (∆t) would be smaller for the jogger in red.

SECTION2.4

How Fast?

• Recall from Chapter 1 that to find the slope, you first choose two points on the line.

• Next, you subtract the vertical coordinate (x in this case) of the first point from the vertical coordinate of the second point to obtain the rise of the line.

• After that, you subtract the horizontal coordinate (t in this case) of the first point from the horizontal coordinate of the second point to obtain the run.

• Finally, you divide the rise by the run to obtain the slope.

Velocity and Speed (cont.)

SECTION2.4

How Fast?

• The slopes of the two lines are found as follows:

• A greater slope, shows that the red jogger traveled faster.

SECTION2.4

How Fast?


• The unit of the slope is meters per second. In other words, the slope tells how many meters the runner moved in 1 s.

• The slope is the change in position, divided by the time interval during which that change took place, or (xf - xi) / (tf - ti), or Δx/Δt.

• When Δx gets larger, the slope gets larger; when Δt gets larger, the slope gets smaller.

SECTION2.4

How Fast?


• The slope of a position-time graph for an object is the object’s average velocity and is represented by the ratio of the change of position to the time interval during which the change occurred.

Average Velocity ≡ _______ = ________ ΔxΔt

(xf - xi)

(tf - ti)

• The symbol ≡ means that the left-hand side of the equation is defined by the right-hand side.

SECTION2.4

How Fast?


• It is a common misconception to say that the slope of a position-time graph gives the speed of the object.

• The slope of the position-time graph on the right is –5.0 m/s. It indicates the average velocity of the object and not its speed.

SECTION2.4

How Fast?


• The object moves in the negative direction at a rate of 5.0 m/s.

SECTION2.4

How Fast?


• The slope’s absolute value is the object’s average speed, 5.0m/s, which is the distance traveled divided by the time taken to travel that distance.

SECTION2.4

How Fast?


• If an object moves in the negative direction, then its displacement is negative. The object’s velocity will always have the same sign as the object’s displacement.

SECTION2.4

How Fast?


The graph describes the motion of a student riding his skateboard along a smooth, pedestrian-free sidewalk. What is his average velocity? What is his average speed?

SECTION2.4

How Fast?


Step 1: Analyze and Sketch the Problem

Identify the coordinate system of the graph.

SECTION2.4

How Fast?



SECTION2.4

How Fast?


Identify the unknown variables.

Unknown:

SECTION2.4

How Fast?


Find the average velocity using two points on the line.

Use magnitudes with signs indicating directions.

= _____ Δx

Δt

(xf - xi)

(tf - ti)

= ______

SECTION2.4

How Fast?


Substitute x2 = 12.0 m, x1 = 6.0 m, t2 = 8.0 s, t1 = 4.0 s:

SECTION2.4

How Fast?


Step 3: Evaluate the Answer

SECTION2.4

How Fast?


Are the units correct?

m/s are the units for both velocity and speed.

Do the signs make sense?

The positive sign for the velocity agrees with the coordinate system. No direction is associated with speed.

SECTION2.4

How Fast?



Step 1: Analyze and Sketch the Problem

Identify the coordinate system of the graph.

SECTION2.4

How Fast?




Find the average velocity using two points on the line.

The average speed is the absolute value of the average velocity.

Step 3: Evaluate the Answer

SECTION2.4

How Fast?


• A motion diagram shows the position of a moving object at the beginning and end of a time interval. During that time interval, the speed of the object could have remained the same, increased, or decreased. All that can be determined from the motion diagram is the average velocity.

• The speed and direction of an object at a particular instant is called the instantaneous velocity.

• The term velocity refers to instantaneous velocity and is represented by the symbol v.

SECTION2.4

How Fast?


• Although the average velocity is in the same direction as displacement, the two quantities are not measured in the same units.

• Nevertheless, they are proportional—when displacement is greater during a given time interval, so is the average velocity.

• A motion diagram is not a precise graph of average velocity, but you can indicate the direction and magnitude of the average velocity on it.

SECTION2.4

How Fast?


Equation of Motion• Using the position-time graph

used before with a slope of -5.0m/s, remember that you can represent any straight line with the equation, y = mx + b.

• y is the quantity plotted on the vertical axis, m is the line’s slope, x is the quantity plotted on the horizontal axis and b is the line’s y-intercept.

SECTION2.4

How Fast?

• Based on the information shown in the table, the equation y = mx + b becomes x = t + xi, or, by inserting the values of the constants, x = (–5.0 m/s)t + 20.0 m.

• You cannot set two items with different units equal to each other in an equation.

Comparison of Straight Lines with Position-Time

Graphs

General Variable

Specific Motion Variable

Value in Graph

y x

m -5.0m/s

x t

b xi 20.0m

SECTION2.4

How Fast?

Equation of Motion (cont.)

• An object’s position is equal to the average velocity multiplied by time plus the initial position.

• This equation gives you another way to represent the motion of an object.

SECTION2.4

How Fast?

Equation of Motion (cont.)

Which of the following statements defines the velocity of the object’s motion?

A. the ratio of the distance covered by an object to the respective time interval

B. the rate at which distance is covered

C. the distance moved by a moving body in unit time

D. the ratio of the displacement of an object to the respective time interval

SECTION2.4

Section Check

Reason: Options A, B, and C define the speed of the object’s motion. The velocity of a moving object is defined as the ratio of the displacement (x) to the time interval (t).

SECTION2.4

Answer

Section Check

Which of the statements given below is correct?

A. Average velocity cannot have a negative value.

B. Average velocity is a scalar quantity.

C. Average velocity is a vector quantity.

D. Average velocity is the absolute value of the slope of a position-time graph.

SECTION2.4

Section Check

Reason: Average velocity is a vector quantity, whereas all other statements are true for scalar quantities.

SECTION2.4

Answer

Section Check

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

A. 2.5 m/s

B. 5 m/s

C. 2 m/s

D. 10 m/s

SECTION2.4

Section Check

Reason: The average velocity of an object is the slope of a position-time graph.

SECTION2.4

Answer

Section Check

Resources

CHAPTER

2

Physics Online

Study Guide

Chapter Assessment Questions

Standardized Test Practice

Representing Motion

• A motion diagram shows the position of an object at successive equal time intervals.

• In the particle model motion diagram, an object’s position at successive times is represented by a series of dots. The spacing between dots indicates whether the object is moving faster or slower.

SECTION2.1

Picturing Motion

Study Guide

• A coordinate system gives the location of the zero point of the variable you are studying and the direction in which the values of the variable increase.

• A vector drawn from the origin of a coordinate system to an object indicates the object’s position in that coordinate system. The directions chosen as positive and negative on the coordinate system.

SECTION2.2

Where and When?

Study Guide

• A time interval is the difference between two times.

• Change in position is displacement, which has both magnitude and direction.

SECTION2.2

Where and When?

Study Guide

• On a motion diagram, the displacement vector’s length represents how far the object was displaced. The vector points in the direction of the displacement, from xi to xf.

SECTION2.2

Where and When?

Study Guide

• Position-time graphs provide information about the motion of objects. They also might indicate where and when two objects meet.

• The line on a position-time graph describes an object’s position at each time.

• Motion can be described using words, motion diagrams, data tables or graphs.

SECTION2.3


Study Guide

• An object’s velocity tells how fast it is moving and in what direction it is moving.

• Speed is the magnitude of velocity.

• Slope on a position-time graph described the average velocity of the object.

SECTION2.4

How Fast?

Study Guide

• You can represent motion with pictures and physical models. A simple equation relates an object’s initial position (xi), its constant average velocity, its position (x) and the time (t) since the object was at its initial position.

SECTION2.4

How Fast?

Study Guide

What should be true about the motion of an object in order for you to treat that object as if it were a particle?

A. The object should be no smaller than your fist.

B. The object should be small compared to its motion.

C. The object should be no larger than you can lift.

D. The object should not be moving faster than the speed of sound.

CHAPTER


Chapter Assessment

Reason: you can treat even planets and stars as particles as long as those objects are small compared to the motion you are studying.

CHAPTER


Chapter Assessment

Which is the distance and direction from one point to another?

A. Displacement

B. Magnitude of distance

C. Position

D. Velocity

CHAPTER


Chapter Assessment

Reason: Velocity is speed and direction.

CHAPTER


Chapter Assessment

On a position-time graph, how would you indicate that object A has a greater velocity than object B?

D. Make the y-intercept for object A greater than the y-intercept for object B.

CHAPTER


Chapter Assessment

A. Make the slope for object A less than the slope for object B.

B. Make the slope for object A greater than the slope for object B.

C. Make the y-intercept for object A less then the y-intercept for object B.

Answer: The slope of a line on a position-time graph indicates the object’s velocity.

CHAPTER


Chapter Assessment

A car is moving at a constant speed of 25 m/s. How far does this car move in 0.2 s, the approximate reaction time for an average person?

A. 5 m

B. 10 m

C. 25 m

D. 50 m

CHAPTER


Chapter Assessment

Reason: (25m/s)(0.2s) = 5m

CHAPTER


Chapter Assessment

Which is a measurement of velocity?

A. 20 m

B. 33 km/s

C. 300 km west

D. 7800 m/s north

CHAPTER


Chapter Assessment

Reason: Velocity measures both speed and direction.

CHAPTER


Chapter Assessment

Which statement about velocity vectors is true?

A. All velocity vectors are positive.

B. Velocity vectors have magnitude but no direction.

C. Velocity vectors and displacement vectors are the same thing.

D. A velocity vector’s length should be proportional to the object’s speed.

CHAPTER



What is the average speed of a sprinter who completes a 55-m dash in 6.2 s?

A. 6.2 m/s

B. 7.1 m/s

C. 8.9 m/s

D. 11 m/s

CHAPTER



Car A is moving faster than Car B on the highway. Which statement describes the particle model motion diagrams for Car A and Car B?

A. The does for Car A are farther apart than the dots for Car B.

B. The dots for Car A are closer together than the dots for Car B.

C. The slope of the motion diagram is greater for Car A than for Car B.

D. The slope of the motion diagram is less for Car A than for Car B.

CHAPTER



An athlete runs four complete laps around a 200-m track. What is the athlete’s displacement?

A. 0 m

B. 200 m

C. 400 m

D. 800 m

CHAPTER



Which correctly describes a relationship between an object’s particle model motion diagram and that object’s graph of position v. time?

A. If the dots on the motion diagram are closer together, then the slope of the graph is greater.

B. If the dots on the motion diagram are farther apart, then the slope of the graph is greater.

C. If the dots on the motion diagram are closer together, then the y-intercept of the graph is less.

D. If the dots on the motion diagram are farther apart, then the y-intercept of the graph is less.

CHAPTER



Stock up on Supplies

Test-Taking Tip

Bring all your test-taking tools: number two pencils, black and blue pens, erasers, correction fluid, a sharpener, a ruler, a calculator, and a protractor.

CHAPTER



Coordinate Systems

CHAPTER


Chapter Resources

Coordinate Systems Showing Position

CHAPTER


Chapter Resources

Motion Diagram Showing Negative Position

CHAPTER


Chapter Resources

Position-Time Graph for the Runner

CHAPTER


Chapter Resources

End of Custom Shows

chapter 2: representing motion physics principles and problems

Documents

picturing motion slide

motion diagrams

objects motion

kinds of motion section

straightline motion

perceiving motion

description of motion

motion physics principles