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Section 2 • Velocity and Momentum Section 3 • Acceleration

How does the slow part of the graph differ from a faster part? What happens on the graph when a cart stops? Why? How does the line look different during negative displacement (going backwards!) ?

Explain what the following distance time graphs tell you about the speed.

Motion Are distance and time important in describing running events at the track-and-field meets in the Olympics? Comstock/JupiterImages

Motion Distance and time are important. In order to win a race, you must cover the distance in the shortest amount of time. How would you describe the motion of the runners in the race? Comstock/JupiterImages

Motion and Position You don't always need to see something move to know that motion has taken place. A reference point is needed to determine the position of an object. Motion occurs when an object changes its position relative to a reference point. The motion of an object depends on the reference point that is chosen.

Distance An important part of describing the motion of an object is to describe how far it has moved, which is distance. The SI unit of length or distance is the meter (m). Longer distances are measured in kilometers (km).

Distance Shorter distances are measured in centimeters (cm).

Displacement Suppose a runner jogs to the 50-m mark and then turns around and runs back to the 20-m mark. The runner travels 50 m in the original direction (north) plus 30 m in the opposite direction (south), so the total distance she ran is 80 m.

Displacement Sometimes you may want to know not only your distance but also your direction from a reference point, such as from the starting point. Displacement is the distance and direction of an object's change in position from the starting point.

Displacement The length of the runner's displacement and the distance traveled would be the same if the runner's motion was in a single direction.

For example:

Adding Displacements Displacements in opposite directions can be subtracted. For example, if you walk 10 m east and then 5 m west, the size of your displacement is:

Adding Displacements Displacements that are not in the same direction or in opposite directions cannot be directly added or subtracted. For example, if you walk 4 m east and then 3 m north, your displacement is 5 m in a roughly northeast direction, but the total distance traveled is 7m.

Speed You could describe movement by the distance traveled and by the displacement from the starting point. You also might want to describe how fast it is moving. Speed is the distance an object travels per unit of time.

Calculating Speed Any change over time is called a rate.
If you think of distance as the change in position, then speed is the rate at which distance is traveled or the rate of change in position.

Calculating Speed The SI unit for distance is the meter and the SI unit of time is the second (s), so in SI, units of speed are measured in meters per second (m/s).

Calculating Speed Sometimes it is more convenient to express speed in other units, such as kilometers per hour (km/h).

What is Speed?

Motion with Constant Speed
Suppose you are in a car traveling on a nearly empty freeway. You look at the speedometer and see that the car's speed hardly changes. If you are traveling at a constant speed, you can measure your speed over any distance interval.

Changing Speed Usually speed is not constant.
Think about riding a bicycle for a distance of 5 km, as shown.

Changing Speed How would you express your speed on such a trip? Would you use your fastest speed, your slowest speed, or some speed between the two?

Average Speed Average speed describes speed of motion when speed is changing. Average speed is the total distance traveled divided by the total time of travel. If the total distance traveled was 5 km and the total time was 1/4 h, or 0.25 h. The average speed was:

“Magic” Circle d t v

Calculating Speed

Instantaneous Speed A speedometer shows how fast a car is going at one point in time or at one instant. The speed shown on a speedometer is the instantaneous speed. Instantaneous speed is the speed at a given point in time. Ryan McGinnis/Getty Images

Instantaneous Speed When something is speeding up or slowing down, its instantaneous speed is changing. If an object is moving with constant speed, the instantaneous speed doesn't change.

Graphing Motion The motion of an object over a period of time can be shown on a distance-time graph. Time is plotted along the horizontal axis of the graph and the distance traveled is plotted along the vertical axis of the graph. Click image to play movie

Plotting a Distance-Time Graph
On a distance-time graph, the distance is plotted on the vertical axis and the time on the horizontal axis. Each axis must have a scale that covers the range of number to be plotted.

Plotting a Distance-Time Graph
Once the scales for each axis are in place, the data points can be plotted. After plotting the data points, draw a line connecting the points.

Section Check Question 1
What is the difference between distance and displacement?

Section Check Answer Distance describes how far an object moves; displacement is the distance and the direction of an object’s change in position.

Section Check Question 2
__________ is the distance an object travels per unit of time. A. acceleration B. displacement C. speed D. velocity

Section Check Answer The answer is C. Speed is the distance an object travels per unit of time.

Section Check Question 3 What is instantaneous speed? Answer
Instantaneous speed is the speed at a given point in time.

Velocity Speed describes only how fast something is moving.
To determine direction you need to know the velocity. Velocity includes the speed of an object and the direction of its motion.

Velocity Because velocity depends on direction as well as speed, the velocity of an object can change even if the speed of the object remains constant. The speed of this car might be constant, but its velocity is not constant because the direction of motion is always changing.

Motion of Earth's Crust As you look around the surface of the Earth from year to year, the basic structure of the planet seems the same. Yet if you examined geological evidence of what Earth's surface looked like over the past 250 million years, you would see that large changes have occurred.

Motion of Earth's Crust Click the play button to see how the continents have moved over time.

Moving Continents These moving plates cause geological changes such as the formation of mountain ranges, earthquakes and volcanic eruptions. The movement of the plates also is changing the size of the oceans and the shapes of the continents.

Relative Motion If you are sitting in a chair reading this sentence, you are moving. You are not moving relative to your desk or your school building, but you are moving relative to the other planets in the solar system and the Sun.

Relative Motion The choice of a reference point influences how you describe the motion of an object. For example, consider a hurricane that is moving towards your house as you evacuate.

Relative Motion If you choose your house as a reference point, the hurricane appears to be approaching at 20 km/h and the car appears to be moving away at 10 km/h.

Relative Motion If you choose your car as a reference point, the hurricane appears to be approaching at 10 km/h and the car appears to be moving away at 10 km/h.

Momentum A moving object has a property called momentum that is related to how much force is needed to change its motion. The momentum of an object is the product of its mass and velocity.

Momentum Momentum is given the symbol p and can be calculated with the following equation: The unit for momentum is kg · m/s. Notice that momentum has a direction because velocity has a direction.

Momentum When two objects have the same velocity, the object with the larger mass has the larger momentum. For example, a 1,500-kg car traveling at 30 m/s east has a momentum of 45,000 kg•m/s east.

Momentum But a 30,000-kg truck traveling at 30 m/s east has a momentum of 900,000 kg•m/s. Furthermore, when two objects have the same mass, the one with the larger velocity has a larger momentum.

Section Check Question 1
A 1,500-kg car is traveling west at 100 m/s. What is the car’s momentum? A. 1,500 kg•m/s B. 150,000 kg•m/s C. 1,400 kg•m/s D. 1,600 kg•m/s

Section Check Answer The answer is B. Momentum is the product of the object’s mass and velocity: 1,500 kg × 100 m/s = 150,000 kg•m/s

Section Check Question 2
The momentum of an object is the product of its __________ and __________. A. mass, acceleration B. mass, velocity C. mass, weight D. net force, velocity

is the product of its mass and velocity, and is given the symbol p.

Section Check Question 3
________ is the speed and direction of an object’s motion. Displacement Motion Velocity Position

Section Check Answer The answer is C. Velocity includes an object’s speed and the direction of its motion.

Acceleration, Speed and Velocity
Acceleration is the rate of change of velocity. When the velocity of an object changes, the object is accelerating. A change in velocity can be either a change in how fast something is moving, or a change in the direction it is moving. Acceleration occurs when an object changes its speed, its direction, or both.

Speeding Up and Slowing Down
When you think of acceleration, you probably think of something speeding up. However, an object that is slowing down also is accelerating. Acceleration also has direction, just as velocity does. A change in velocity can be either a change in how fast something is moving or a change in the direction of movement.

Speeding Up and Slowing Down
If the acceleration is in the same direction as the velocity, the speed increases.

Speeding Up and Slowing Down
If the speed decreases, the acceleration is in the opposite direction from the velocity, and the acceleration is negative.

Changing Direction Any time a moving object changes direction, its velocity changes and it is accelerating.

Speed-time Graphs For objects traveling in a straight line, a speed-time graph can provide information about the object’s acceleration. The slope of the line on a speed-time graph equals the object’s acceleration.

Calculating Acceleration
To calculate the acceleration of an object, the change in velocity is divided by the length of time interval over which the change occurred. To calculate the change in velocity, subtract the initial velocity—the velocity at the beginning of the time interval—from the final velocity—the velocity at the end of the time interval.

Calculating Acceleration
Then the change in velocity is:

Calculating Acceleration
Using this expression for the change in velocity, the acceleration can be calculated from the following equation:

Calculating Acceleration
If the direction of motion doesn't change and the object moves in a straight line, the size of change in velocity is the same as the change in speed. The size of change in velocity then is the final speed minus the initial speed.

Speeding Up Suppose a jet airliner starts at rest at the end of a runway and reaches a velocity of 80 m/s east in 20 s. David Frazier/Corbis

Speeding Up The airliner is traveling in a straight line down the runway, so its speed and velocity are the same size. Because it started from rest, its initial speed was zero.

Speeding Up Its acceleration can be calculated as follows:

Slowing Down Now imagine that a skateboarder is moving in a straight line with a velocity of 3 m/s and north comes to a stop in 2 s. The final speed is zero and the initial speed was 3 m/s. Ken Karp for MMH

Calculating Negative Acceleration
The skateboarder's acceleration is calculated as follows: The acceleration is in the opposite direction of the skateboard’s velocity when the skateboarder is slowing down.

Motion in Two Dimensions
Most objects do not move in only in straight lines. When an object changes direction, it is accelerating. Like displacement and velocity, accelerations in the same direction can be added and accelerations in opposite directions can be subtracted. Accelerations that are not in the same direction or in opposite directions cannot be directly added together.

Changing Direction The speed of the horses in this carousel is constant, but the horses are accelerating because their direction is changing constantly.

Circular Motion When a ball enters a curve, even if its speed does not change, it is accelerating because its direction is changing. When a ball goes around a curve, the change in the direction of the velocity is toward the center of the curve.

Circular Motion Acceleration toward the center of a curved or circular path is called centripetal acceleration.

Projectile Motion If you’ve tossed a ball to someone, you’ve probably noticed that thrown objects don’t always travel in straight lines. They curve downward. Earth’s gravity causes projectiles to follow a curved path.

Horizontal and Vertical Motions
When you throw a ball, the force exerted by your hand pushes the ball forward. This force gives the ball horizontal motion. No force accelerates it forward, so its horizontal velocity is constant, if you ignore air resistance. Donald Miralle/Getty Images

Horizontal and Vertical Motions
However, when you let go of the ball, gravity can pull it downward, giving it vertical motion. The ball has constant horizontal velocity but increasing vertical velocity.

Horizontal and Vertical Motions
Gravity exerts an unbalanced force on the ball, changing the direction of its path from only forward to forward and downward. The result of these two motions is that the ball appears to travel in a curve.

Click image to view movie
Horizontal and Vertical Distance If you were to throw a ball as hard as you could from shoulder height in a perfectly horizontal direction, would it take longer to reach the ground than if you dropped a ball from the same height? Click image to view movie

Horizontal and Vertical Distance
Surprisingly, it wouldn’t. Both balls travel the same vertical distance in the same amount of time.

Amusement Park Acceleration
Engineers use the laws of physics to design amusement park rides that are thrilling, but harmless. The highest speeds and accelerations usually are produced on steel roller coasters. CORBIS

Amusement Park Acceleration
Steel roller coasters can offer multiple steep drops and inversion loops, which give the rider large accelerations. As the rider moves down a steep hill or an inversion loop, he or she will accelerate toward the ground due to gravity.

Amusement Park Acceleration
When riders go around a sharp turn, they also are accelerated. This acceleration makes them feel as if a force is pushing them toward the side of the car.

Section Check Question 1
Acceleration is the rate of change of __________.

Section Check Answer The correct answer is velocity. Acceleration occurs when an object changes its speed, direction, or both.

Section Check Question 2 Which is NOT a form of acceleration?
A. maintaining a constant speed and direction B. speeding up C. slowing down D. turning

Section Check Answer The answer is A. Any change of speed or direction results in acceleration.

Section Check Question 3
What is the acceleration of a hockey player who is skating at 10 m/s east and comes to a complete stop in 2 s? A. 5 m/s2 east B. 5 m/s2 west C. 20 m/s2 east D. 20 m/s2 west

Section Check Answer The answer is B. Calculate acceleration by subtracting initial velocity (10 m/s) from final velocity (0), then dividing by the time interval (2s).

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End of Chapter Summary File

Chapter Resources Click on one of the following icons to go to that resource. connected.mcgraw-hill.com/ Image Bank Video Clips and Animations Chapter Summary Chapter Review Questions Standardized Test Practice

Image Bank Click on individual thumbnail images to view larger versions.

Image Bank Athlete Running Comstock/JupiterImages

Image Bank Conversion Table

Image Bank Distance vs. Displacement

Image Bank Units of Speed

Image Bank Displacement

Image Bank Speed Changing over Distance

Image Bank Speedometer Ryan McGinnis/Getty Images

Image Bank Graphing Motion

Image Bank Positive Acceleration

Image Bank Negative Acceleration

Image Bank Changing Direction

Image Bank Changing Direction

Image Bank Positive Acceleration David Frazier/Corbis

Image Bank Negative Acceleration Ken Karp for MMH

Image Bank Amusement Park Acceleration CORBIS

Image Bank Circular Motion

Image Bank Pitcher Donald Miralle/Getty Images

Image Bank Speed-Time Graph

Image Bank Relative Motion

Video Clips and Animations
Click image to play movie Click here to view the next video clip.

Video Clips and Animations

Reviewing Main Ideas Describing Motion
Motion is a change of position of a body. Distance is the measure of how far an object moved. Displacement is the distance and direction of an object's change in position from the starting point. A reference point must be specified in order to determine an object's position.

Reviewing Main Ideas Describing Motion
The speed of an object can be calculated from this equation: The slope of a line on a distance-time graph is equal to the speed. Velocity describes the speed and direction of a moving object.

Reviewing Main Ideas Velocity and Momentum
Velocity describes the speed and direction of a moving object. Momentum is the product of an object’s mass and velocity.

Reviewing Main Ideas Acceleration
Acceleration occurs when an object changes speed or changes direction. An object speeds up if its acceleration is in the direction of its motion. An object slows down if its acceleration is opposite to the direction of its motion. Acceleration is the rate of change of velocity, and is calculated from this equation:

Chapter Review Question 1
Which includes both the speed of a moving object and the direction of its motion? A. displacement B. distance C. instantaneous speed D. velocity

Chapter Review Question 2
What is the average speed in m/s of a runner who travels 10.0 km in 1.5 hours? A m/s B m/s C m/s D m/s

Chapter Review Question 3
A car travels at a constant speed of 40 m/s for 2 hours. What was the distance traveled in km?

Chapter Review Answer The car traveled 288 km. There are 3,600 s in 1 h and 1,000 m in 1 km. Step 1: 40 m/s × 7,200 = 288,000 Step 2: 288,000= 288 km 1,000

Standardized Test Practice
Question 1 A moving object traveled 4900 m in 165 s. What is its average speed? A m/s B. 55 m/s C. 30 m/s D. 15 m/s

Standardized Test Practice
Answer The answer is C. Average speed is equal to the total distance traveled divided by the total time of travel.

Standardized Test Practice
Question 2 This graph represents swim times for three different swimmers. Which swimmer did not maintain a constant speed?

Standardized Test Practice
A. A B. B C. C D. All speeds were constant.

Standardized Test Practice
Answer The answer is C. During the time interval between 10 and 20 minutes, swimmer C had a speed of 0 m/min.

Standardized Test Practice
Question 3 Calculate the acceleration of a car that starts at rest and has a final velocity of 30 m/s north after 10 s. A. 3 m/s2 South B. 3 m/s2 North C. 20 m/s2 South D. 20 m/s2 North

Standardized Test Practice
Answer The answer is B. Use this equation to calculate acceleration: a = (vf – vi)/t

Standardized Test Practice
Question 4 Car A is traveling north at 85 km/h. After 1 hour, the car turns around 180º and travels at the same speed for 1.5 hours. What is the car’s displacement at the end of the trip? A km South B km South C km South D km North

Standardized Test Practice
Answer The answer is A, 42.5 km South. During the first hour the car traveled 85 km North. During the second hour, the car traveled km South. Even though the total distance traveled is km, the car is 42.5 km South of its starting point.

Standardized Test Practice
Question 5 What is Bill’s average speed? Runner Distance (km) Time (min) Carlos 14.5 42 Bill 8.0 38 Janet 12.4 32 Ann 10.3 30

Standardized Test Practice
A km/min B km/min C km/min D km/min Runner Distance (km) Time (min) Carlos 14.5 42 Bill 8.0 38 Janet 12.4 32 Ann 10.3 30

Standardized Test Practice