1. Chapter 2 Motion in One Dimension

2. Section 2-1: Displacement & Velocity One-dimensional motion is the simplest form of motion. One way to simplify the concept of motion is to consider only the kinds of motion that takes place in one dimension. An example of one-dimensional motion is the motion of a commuter train on a straight track. In this one-dimensional motion, the train can move either forward or backward along the tracks.

3. It can’t move left and right or up and down. Later on, we will learn how to describe more complicated motions by breaking them down into examples of one- dimensional motion.

4. Motion takes place over time and depends upon the frame of reference. It seems simple to describe the motion of the train. As the train begins its route, it is at the first station. Later, it will be at another station farther down the tracks. But what about all of the motions around the train?

5. The earth is spinning on its axis, so the train, stations, and the tracks are also moving around the axis. At the same time, the earth is moving around the sun. The sun and the rest of the solar system are moving through our galaxy. The galaxy is traveling through space as well. Whenever faced with a complex situation like this, physicists break it down into simpler parts.

6. One key approach is to choose a frame of reference against which you can measure changes in position. Frame of reference-- a coordinate system for specifying the precise location of objects in space. We pick a stationary reference object to compare the position of the object in question to. In the case of the train, the stations, along its route are convenient frames of reference. If an object is at rest, its position does not change with respect to a frame of reference.

7. In physics, any frame of reference can be chosen as long as it is used consistently. Our common reference object is the surface of the earth which we consider to be stationary. Motion– the displacement of an object in relation to objects that are considered to be stationary.

8. Displacement As any object moves from one position to another, the length of the straight line drawn from its initial position to the object’s final position is called the displacement of the object. Displacement is a change in position.

9. The gecko in figure 2-2 moves from left to right along the x-axis from an initial position, x i, to a final position, x f. The gecko’s displacement is the difference between its final and initial coordinates, or x f – x i. In this case, the displacement is about 63 cm (85 cm – 22 cm). The formula for displacement: Δx = x f –x i The Greek letter delta means “change in.”

10. Now, suppose the gecko runs up a tree. In this case, we place the measuring stick parallel to the tree. The measuring stick can serve as the y-axis of our coordinate system. The gecko’s initial and final positions are indicated by y i and y f. The gecko’s displacement will be denoted as Δy.

11. Displacement is not always equal to the distance traveled. Displacement is always a straight line segment from one point to another point, even though the actual path of the moving object between 2 points might be a curve. Displacement can be positive or negative. Displacement also includes a description of motion. In one-dimensional motion, there are only two directions in which an object can move, and these directions can be described as positive and negative.

12. In this book, right will be considered the positive direction and the left will be considered the negative direction. Upward will be considered positive and downward will be considered negative.

14. Velocity Knowing speed is important when evaluating motion. Average velocity, v avg, is defined as the displacement divided by the time interval during which the displacement occurred. The SI unit for velocity is m/s. The formula for average velocity is: v = Δx/Δt = (x f – x i )/(t f – t i )

15. The average velocity of an object can be positive or negative, depending on the sign of the displacement. Remember, the time interval must always be positive.

16. Velocity is not the same as speed. In everyday language, the terms speed and velocity are used interchangeably. Velocity describes motion with both a direction and a magnitude. Speed has no direction, only magnitude. Average speed is equal to the distance traveled divided by the time interval for the motion.

17. Velocity can be interpreted graphically. The velocity of an object can be determined if its position is known at specific times along its path. One way to determine this is to make a graph of the motion. Time is plotted on the x-axis and position is plotted on the vertical axis. Refer to page 45 in your text book. An object moving with constant velocity is represented by a straight line on the position-time graph.

18. For any position-time graph, the average velocity can be determined by drawing a straight line between any two points on the graph. The slope of this line indicates the average velocity between the positions and times represented by these points. To understand why this is so, compare the equation for the slope of the line with the equation for average velocity. Figure 2-6 on page 46 represents straight-line graphs of position versus time for three different objects.

19. Instantaneous velocity is the velocity of an object at some instant (or specific point in its path.) Instantaneous velocity may not be the same as average velocity.

20. Section 2-2: Acceleration The gas pedal of a car is sometimes called the accelerator. When you drive a car along a level highway and increase the pressure on the accelerator, the speed of the car increases. We say that the car is accelerating.

21. When you apply the brakes of the car that you are driving, the speed of the car decreases. The speedometer indicates that the car’s speed is decreasing, and we say that the car is decelerating. Acceleration– the rate of change of velocity. The formula for average acceleration: a avg = Δv/Δt = (v f – v i )/(t f –t i ) The SI unit for acceleration is m/s 2.

22. Acceleration has direction and magnitude. Imagine that a train is moving to the right, so that the displacement and the velocity are positive. The velocity increases in magnitude as the train picks up speed. The final velocity will be greater than the initial velocity, and Δv will be positive. When Δv is positive, the acceleration is positive.

23. On long trips with no stops, the train may travel for a while at a constant velocity. In this situation, because the velocity is not changing, Δv = 0 m/s. When velocity is constant, the acceleration is equal to zero. Imagine that the train, still traveling in the positive direction, slows down as it approaches the next station. In this case, the velocity is still positive, but the acceleration is negative because the initial velocity is larger than the final velocity, so Δv will be negative.

24. The slope and shape of a graph describe the object’s motion. A graph of acceleration can be obtained by plotting velocity versus time.

26. This table shows how the signs of the velocity and acceleration can be combined to give a description of the object’s motion.

27. Motion with Constant Acceleration The following picture is a strobe photograph of a ball moving in a straight line with constant acceleration. While the ball was moving, its image was captured ten times in one second, so the time interval between successive images is 0.10 s. As the ball’s velocity increases, the ball travels a greater distance during each time interval.

29. Because the acceleration is constant, the velocity increases by the same amount in each time interval. Because the velocity increases by the same amount in each time interval, the displacement for each time interval increases by the same amount. In other words, the distance that the ball travels in each time interval is equal to the distance it traveled in the previous time interval, plus a constant distance. The relationships between displacement, velocity, and constant acceleration are expressed by equations that apply to any object moving with constant acceleration.

30. Section 2-3: Falling Objects Freely falling bodies undergo constant acceleration. It is now well known that in the absence of air resistance all objects dropped near the surface of a planet fall with the same constant acceleration. Free fall– motion of an object falling with a constant acceleration.

32. In the previous figure, a feather and an apple are released from rest in a vacuum chamber. The trapdoor that released the two objects was opened with an electronic switch at the same instant the camera shutter was opened. The two objects fell at exactly the same rate, as indicated by the horizontal alignment of the multiple images.

33. The amount of time that passed between the first and second images is equal to the amount of time that passed between the fifth and sixth images. However, the picture shows that the displacement in each time interval did not remain constant. Because of this, the velocity is not constant. The feather and the apple were accelerating.

34. Free fall acceleration is denoted with the symbol g. At the surface of Earth, the magnitude of g is approximately 9.81 m/s 2. Unless otherwise stated, the book will use this value for calculations. This acceleration is directed downward, toward the center of the Earth. In our usual choice of coordinates, the downward direction is negative. The acceleration of objects in free fall near the surface of the Earth is g = -9.81 m/s 2.

35. The following figure is a strobe photograph of a ball thrown up into the air with an initial upward velocity of +10.5 m/s. The photo on the left shows the ball moving up from its release to the top of its path, and the photo on the right shows the ball moving down from the top of its path. Everyday experience shows that when we throw an object up in the air, it will continue to move upward for some time, stop momentarily at the peak, and then change direction and begin to fall.

37. Because the object changes direction, it may seem like the velocity and direction are both changing. Actually, objects thrown into the air have a downward acceleration as soon as they are released. In the photo on the left, the upward displacement of the ball between each successive image is smaller and smaller until the ball stops and finally begins to move with an increasing downward velocity, as shown on the right.

38. As soon as the ball is released with an initial upward velocity of +10.5 m/s, it has an acceleration of –9.81 m/s 2. After 1.0 s, the ball’s velocity will change by –9.81 m/s to 0.69m/s upward. After 2.0 s, the ball’s velocity will again change by –9.81 m/s, to –9.12 m/s. The following graph shows the velocity of the ball plotted against time. As you can see, there is an instant when the velocity of the ball is equal to 0 m/s.

40. The velocity is zero at the instant the ball reaches the peak of its upward motion and is about to begin moving downward. Although the velocity is zero at the instant the ball reaches the peak, the acceleration is equal to –9.81 m/s 2 at every instant regardless of the magnitude or direction of the velocity. It is important to note that the acceleration is –9.81 m/s 2 even at the peak where the velocity is zero. The straight line slope of the graph indicates that the acceleration is constant at every moment.

41. It may seem a little confusing to think of something that is moving upward as having a downward acceleration. Thinking of this motion as motion with a positive velocity and a negative acceleration may help. The downward acceleration is the same when an object is moving up, when it is at rest at the top of its path, and when it is moving down. The only things changing are the position and the magnitude and direction of the velocity.

42. Knowing the free-fall acceleration makes it easy to calculate the velocity, time, and displacement of many different motions using the equations for constantly accelerated motion. Because the acceleration is the same throughout the entire motion, you can analyze the motion of a freely-falling object during any time interval.