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Chapter 11 Motion

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**Chapter 11 Sections 11.1 Distance & Displacement 11.2 Speed & Velocity**

11.3 Acceleration

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**11.1 Distance & Displacement Key Concepts**

What is needed to describe motion completely? How are distance and displacement different? How do you add displacements?

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11.1 Describing Motion To describe motion, you must state the direction the object is moving as well as how fast the object is moving. You must also tell its location at a certain time. To describe motion accurately and completely, a frame of reference is necessary. The necessary ingredient of a description of motion—a frame of reference—is a system of objects that are not moving with respect to one another.

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11.1 Describing Motion Relative motion is movement in relation to a frame of reference. As a train moves past a platform, people standing on the platform will see those on the train speeding by. When the people on the train look at one another, they don't seem to be moving at all. Choosing a meaningful frame of reference allows you to describe motion in a clear and relevant manner.

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11.1 Measuring Distances Distance is the length of a path between two points. When an object moves in a straight line, the distance is the length of the line connecting the object's starting point and its ending point. The SI unit for measuring distance is the meter (m). For very large distances, it is more common to make measurements in kilometers (km). One kilometer equals 1000 meters. Distances that are smaller than a meter are measured in centimeters (cm). One centimeter is one hundredth of a meter.

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**11.1 Measuring Displacements**

To describe an object's position relative to a given point, you need to know how far away and in what direction the object is from that point. Distance is the length of the path between two points. Displacement is the direction from the starting point and the length of a straight line from the starting point to the ending point.

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**11.1 Measuring Displacements**

If you measure the path along which a roller coaster car has traveled, you are describing distance. The direction from the starting point to the car and the length of the straight line from the starting point to the car describe displacement. After completing one trip around the track, the roller coaster car's displacement is zero.

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**11.1 Vectors Displacement is an example of a vector.**

A vector is a quantity that has magnitude and direction. The magnitude can be size, length, or amount. Arrows on a graph or map are used to represent vectors. The length of the arrow shows the magnitude of the vector. Vector addition is the combining of vector magnitudes and directions. Add displacements using vector addition.

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11.1 Vectors When two displacements, represented by two vectors, have the same direction, you can add their magnitudes. In Figure 3A, the total magnitude of the displacement is 6 kilometers. If two displacements are in opposite directions, the magnitudes subtract from each other, as shown in Figure 3B. Because the car's displacements (4 kilometers and 2 kilometers) are in opposite directions, the magnitude of the total displacement is 2 kilometers.

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11.1 Vectors When two or more displacement vectors have different directions, they may be combined by graphing. Figure 4 shows vectors representing the movement of a boy walking from his home to school. The lengths of the vectors representing this path are 1 block, 1 block, 2 blocks, and 3 blocks. You can determine this distance by adding the magnitudes of each vector along his path.

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11.1 Vectors The vector in red is called the resultant vector, which is the vector sum of two or more vectors. In this case, it shows the displacement. The resultant vector points directly from the starting point to the ending point. Vector addition, then, shows that the boy's displacement is 5 blocks approximately northeast, while the distance he walked is 7 blocks.

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**11.2 Speed & Velocity Key Concepts**

How are instantaneous speed and average speed different? How can you find the speed from a distance time graph? How are speed and velocity different? How do velocities add?

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11.2 Speed Speed is the ratio of the distance an object moves to the amount of time the object moves. The SI unit of speed is meters per second (m/s). You need to choose units that make the most sense for the motion you are describing. The in-line skater in Figure 5 may travel 2 meters in one second. The speed would be expressed as 2 m/s. A car might travel 80 kilometers in one hour. Its speed would be expressed as 80 km/h.

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11.2 Average Speed Two ways to express the speed of an object are average speed and instantaneous speed. Average speed is computed for the entire duration of a trip, and instantaneous speed is measured at a particular instant. Sometimes it is useful to know how fast something moves for an entire trip. Average speed, , is the total distance traveled, d, divided by the time, t, it takes to travel that distance.

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11.2 Instantaneous Speed Sometimes however, such as when driving on the highway, you need to know how fast you are going at a particular moment. The car's speedometer gives your instantaneous speed. Instantaneous speed, v, is the rate at which an object is moving at a given moment in time. For example, you could describe the instantaneous speed of the car in Figure 6 as 55 km/h.

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11.2 Graphing Motion A distance-time graph is a good way to describe motion. Recall that slope is the change in the vertical axis value divided by the change in the horizontal axis value. The slope of a line on a distance-time graph is speed. In Figure 7A, the car travels 25.0 meters per second. In Figure 7B the slope of the line is 250.0 meters divided by 20.0 seconds, or 12.5 meters per second. A steeper slope on a distance-time graph indicates a higher speed.

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11.2 Graphing Motion Figure 7C shows the motion of a car that is not traveling at a constant speed. The times when the car is gradually increasing or decreasing its speed are shown by the curved parts of the line. The slope of the straight portions of the line represent periods of constant speed. Note that the car's speed is 25 meters per second during the first part of its trip and 38 meters per second during the last part of its trip.

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11.2 Velocity Sometimes knowing only the speed of an object isn't enough. You also need to know the direction of the object's motion. Together, the speed and direction in which an object is moving are called velocity. Velocity is a description of both speed and direction of motion. Velocity is a vector. A longer vector would represent a faster speed, and a shorter one would show a slower speed. The vectors would also point in different directions to represent the cheetah's direction at any moment.

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11.2 Velocity A change in velocity can be the result of a change in speed, a change in direction, or both. The sailboat can be described as moving with uniform motion, which is another way of saying it has constant velocity. However, the sailboat's velocity also changes if it changes its direction. It may continue to move at a constant speed, but the change of direction is a change in velocity.

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11.2 Combining Velocities Sometimes the motion of an object involves more than one velocity. Two or more velocities add by vector addition. The velocity of the river relative to the riverbank (X) and the velocity of the boat relative to the river (Y) combine. They yield the velocity of the boat relative to the riverbank (Z). This velocity is 17 kilometers per hour downstream.

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11.2 Combining Velocities The relative velocities of the current (X) and the boat (Y) are at right angles to each other. Adding these velocity vectors yields a resultant velocity of the boat relative to the riverbank of 13 km/h (Z). Note that this velocity is at an angle to the riverbank and forms a right triangle.

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**11.3 Acceleration Key Concepts**

How are changes in velocity described? How can you calculate acceleration? How does a speed time graph indicate acceleration? What is instantaneous acceleration?

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11.3 Acceleration The rate at which velocity changes is called acceleration. Recall that velocity is a combination of speed and direction. Acceleration can be described as changes in speed, changes in direction, or changes in both. Acceleration is a vector.

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11.3 Acceleration We often use the word acceleration to describe situations in which the speed of an object is increasing. Scientifically, however, acceleration applies to any change in an object's velocity. This change may be either an increase or a decrease in speed. Acceleration can be caused by positive (increasing) change in speed or by negative (decreasing) change in speed.

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**11.3 Acceleration Due To Gravity**

An example of acceleration due to change in speed is free fall, the movement of an object toward Earth solely because of gravity. The unit for acceleration is meters per second per second. This unit is typically written as meters per second squared (m/s2). Objects falling near Earth's surface accelerate downward at a rate of 9.8 m/s2.

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**11.3 Acceleration From Changing Direction**

You can accelerate even if your speed is constant. You have experienced this type of acceleration if you have ridden on a carousel like the one in Figure 13. A horse on the carousel is traveling at a constant speed, but it is accelerating because its direction is constantly changing.

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**11.3 Changing Speed & Direction**

Sometimes motion changes in both speed and direction at the same time. You experience this type of motion if you ride on a roller coaster. You are thrown backward, forward, and sideways as your velocity increases, decreases, and changes direction. Your acceleration is constantly changing because of changes in the speed and direction of the cars of the roller coaster.

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**11.3 Constant Acceleration**

Constant acceleration is a steady change in velocity. The velocity of the object changes by the same amount each second. An example of constant acceleration is illustrated by the jet airplane shown in Figure 15. The airplane's acceleration may be constant during a portion of its takeoff.

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**11.3 Calculating Acceleration**

Acceleration is the rate at which velocity changes. You calculate acceleration for straight-line motion by dividing the change in velocity by the total time. If a is the acceleration, vi is the initial velocity, vf is the final velocity, and t is total time, this equation can be written as follows.

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**11.3 Graphing Acceleration**

You can use a graph to calculate acceleration. Figure 16 is a graph of the skier's speed. The slope of a speed-time graph is acceleration. This slope is change in speed divided by change in time.

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11.3 Speed-Time Graphs Constant acceleration is represented on a speed–time graph by a straight line. The graph in Figure 16 is an example of a linear graph, in which the displayed data form straight-line parts. The slope of the line is the acceleration.

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**11.3 Speed-Time Graphs Constant negative acceleration decreases speed.**

A speed-time graph of the motion of a bicycle slowing to a stop is shown in Figure 17. The line segment sloping downward represents the bicycle slowing down. The change in speed is negative, so the slope of the line is negative.

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11.3 Distance-Time Graphs Accelerated motion is represented by a curved line on a distance-time graph. In a nonlinear graph, a curve connects the data points that are plotted. Figure 18 is a distance-time graph. Notice that the slope is much greater during the fourth second than it is during the first second. Because the slope represents the speed of the ball, an increasing slope means that the ball is accelerating.

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**11.3 Instantaneous Acceleration**

Acceleration is rarely constant, and motion is rarely in a straight line. A skateboarder moving along a half-pipe changes speed and direction. At each moment she is accelerating, but her instantaneous acceleration is always changing. Instantaneous acceleration is how fast a velocity is changing at a specific instant.

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**11.3 Acceleration As A Vector**

Acceleration involves a change in velocity or direction or both, so the vector of the skateboarder's acceleration can point in any direction. The vector's length depends on how fast she is changing her velocity. At every moment she has an instantaneous acceleration, even if she is standing still and the acceleration vector is zero.

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