Accelerated MotionCHAPTER 3

Acceleration is the rate of change in an objects velocity.

SECTIONSSECTIONS

WATCH THIS!C S VideoSKATEBOARD PHYSICS How does a trip to your local skate park involve physics? You might be surprised! Explore acceleration as skateboarders show off their best moves.

LaunchLAB iLab StationGRAPHING MOTION How does a graph showing constant speed compare to a graph of an object that is accelerating?

1 Acceleration

2 Motion with Constant Acceleration

3 Free Fall

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Nonuniform Motion Diagrams An object in uniform motion moves along a straight line with an

unchanging velocity, but few objects move this way all the time. More common is nonuniform motion, in which velocity is changing. In this chapter, you will study nonuniform motion along a straight line. Exam-ples include balls rolling down hills, cars braking to a stop, and falling objects. In later chapters you will analyze nonuniform motion that is not confined to a straight line, such as motion along a circular path and the motion of thrown objects, such as baseballs.

Describing nonuniform motion You can feel a difference between uniform and nonuniform motion. Uniform motion feels smooth. If you close your eyes, it feels as if you are not moving at all. In contrast, when you move around a curve or up and down a roller coaster hill, you feel pushed or pulled.

How would you describe the motion of the person in Figure 1? In the first diagram, the person is motionless, but in the others, her position is changing in different ways. What information do the diagrams contain that could be used to distinguish the different types of motion? Notice the distances between successive positions. Because there is only one image of the person in the first diagram, you can conclude that she is at rest. The distances between images in the second diagram are the same because the jogger is in uniform motion; she moves at a constant veloc-ity. In the remaining two diagrams, the distance between successive positions changes. The change in distance increases if the jogger speeds up. The change decreases if the jogger slows down.

An object accelerates when its velocity changesthat is, when it speeds up, slows down, or changes direction.

Essential Questions What is acceleration?

How is acceleration different from velocity?

What information can you learn from velocity-time graphs?

Review Vocabularyvector a quantity that has magnitude and direction

New Vocabularyaccelerationvelocity-time graphaverage accelerationinstantaneous acceleration

As an airplane takes off, its speed changes from 5 m/s on the runway to nearly 300 m/s once its in the air. If youve ever ridden on an airplane, youve felt the seat push against your back as the plane rapidly accelerates.

SECTION 1 AccelerationAcceleration

a. The person is motionless. b. Equally spaced images show her moving at a constant speed.

c. She is speeding up. d. She is slowing down.

Figure 1 The distance the jogger moves in each time interval indicates the type of motion.

n Motion Diagram

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First, draw vf. Below that,draw vi with its tail alignedwith the tip of vf.

Next, draw the vector v fromthe tail of vf to the tip of vi. Theacceleration vector a is the sameas v divided by the time interval.

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Finding Acceleration Vectors

Particle model diagram What does a particle model motion diagram look like for an object with changing velocity? Figure 2 shows particle model motion diagrams below the motion diagrams of the jogger when she is speeding up and slowing down. There are two major indicators of the change in velocity in this form of the motion diagram. The change in the spacing of the dots and the differences in the lengths of the velocity vectors indicate the changes in velocity. If an object speeds up, each subsequent velocity vector is longer, and the spacing between dots increases. If the object slows down, each vector is shorter than the previous one, and the spacing between dots decreases. Both types of motion diagrams indicate how an objects velocity is changing.

READING CHECK Analyze What do increasing and decreasing lengths of velocity vectors indicate on a motion diagram?

Displaying acceleration on a motion diagram For a motion diagram to give a full picture of an objects movement, it should contain information about the rate at which the objects velocity is changing. The rate at which an objects velocity changes is called the acceleration of the object. By including acceleration vectors on a motion diagram, you can indicate the rate of change for the velocity.

Figure 3 shows a particle motion diagram for an object with increasing velocity. Notice that the lengths of the red velocity vectors get longer from left to right along the diagram. The figure also describes how to use the diagram to draw an acceleration vector for the motion. The acceleration vector that describes the increasing velocity is shown in violet on the diagram.

Notice in the figure that if the object's acceleration is constant, you can determine the length and direction of an acceleration vector by subtracting two consecutive velocity vectors and dividing by the time interval. That is, first find the change in velocity, v = vf vi = vf + (vi), where vi and vf refer to the velocities at the beginning and the end of the chosen time interval. Then divide by the time interval (t). The time interval between each dot in Figure 3 is 1 s. You can draw the acceleration vector from the tail of the final velocity vector to the tip of the initial velocity vector.

Figu re 2 The change in length of the veloc-ity vectors on these motion diagrams indi-cates whether the jogger is speeding up or slowing down.

Figure 3 For constant acceleration, an acceleration vector on a particle model dia-gram is the difference in the two velocity

vectors divided by the time interval: a = v _ t

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Analyze Can you draw an acceleration vector for two successive velocity vectors that are the same length and direction? Explain.

of vfvv to the tip of vi. Theration vector a is the samedivided by the time interval.

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Direction of AccelerationConsider the four situations shown in Figure 4 in which an object can

accelerate by changing speed. The first motion diagram shows the car moving in the positive direction and speeding up. The second motion diagram shows the car moving in the positive direction and slowing down. The third shows the car speeding up in the negative direction, and the fourth shows the car slowing down as it moves in the negative direction. The figure also shows the velocity vectors for the second time interval of each diagram, along with the corresponding acceleration vectors. Note that t is equal to 1 s.

In the first and third situations, when the car is speeding up, the velocity and acceleration vectors point in the same direction. In the other two situations, in which the acceleration vector is in the opposite direction from the velocity vectors, the car is slowing down. In other words, when the car's acceleration is in the same direction as its velocity, the car's speed increases. When they are in opposite directions, the speed of the car decreases.

Both the direction of an objects velocity and its direction of accelera-tion are needed to determine whether it is speeding up or slowing down. An object has a positive acceleration when the acceleration vector points in the positive direction and a negative acceleration when the accelera-tion vector points in the negative direction. It is important to notice that the sign of acceleration alone does not indicate whether the object is speeding up or slowing down.

READING CHECK Describe the motion of an object if its velocity and accel-eration vectors have opposite signs.

Figure 4 You need to know the direction of both the velocity and acceleration vectors in order to determine whether an object is speeding up or slowing down.

Investigate accelerated motion.

Virtual Investigation

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