Working through this chapter of the study guide will enable you to:

1. Decide when to use vectors and when to use scalars when describing the motion of an object.
2. Learn about the physical processes and theories that are involved in the study of motion.
3. Identify the units in which each of the concepts of motion is expressed.
4. Distinguish among straight-line motion, uniform circular motion, and projectile motion.
5. Discover how acceleration due to gravity affects projectile motion near Earth's surface.

Discussion

One of the first things we notice about objects is whether or not they are in motion. This probably results from the fact that moving objects are often dangerous, and we must be ready to avoid those that present a threat to our safety. Even non-threatening things are usually much more interesting if they are moving than if they are at rest. Think about looking at a picture of a horse in comparison to the actual experience of watching that horse jump in an international horse show on television. Better yet, imagine the thrill of attending a live jumping competition, where not only the motion of the horse is evident, but you may also experience the weather, the excitement of the crowd, and even the sensations of smell, touch, and taste that are related to a horse show.


Section   2.1Defining Motion

Any object that is changing its position is said to be in motion. Studying the motion of objects involves the basic techniques of measurement that we learned in the previous chapter. The only units needed are length and time, so at first glance you might expect this study to be quite simple. As you will see, however, limiting our measurements to just two of the fundamental units does not make the mathematics and calculations all that trivial. As is the case in most situations, the keys to understanding this chapter are the definitions of the terms used to describe motion. In this section the most important terms are distance and displacement. Distance is the actual path length traveled by a moving object. Displacement is the straight-line separation between the starting and ending positions for the segment of motion under study. In addition, displacement has a specific direction associated with it, which means that it must be treated as a vector quantity.


Section   2.2Speed and Velocity

Once the important distinction between distance and displacement has been learned, the concepts of speed and velocity can be defined by using the interaction of time with changes in position. The scalar quantity speed deals with the change in distance divided by the time and requires only magnitude (a number and the related units) when it is specified. Velocity is defined as displacement divided by the time and must have magnitude plus direction. This means that velocity must be treated as a vector quantity, just as is displacement. As a simple example, a person might walk along a winding path through the woods for a total distance of 83 meters. When he looks around, he finds that he is now only 60 meters directly north of his starting point. If the trip took 6.0 minutes (360 seconds), his speed of travel can be calculated as the distance divided by the time, 83 meters/360 seconds, or 0.23 meters/second. On the other hand, to calculate his velocity we must take his displacement and divide it by the time. This gives 60 meters divided by 360 seconds or 0.17 meters/second, due north. The specification of the direction north is necessary because velocity is a vector quantity and is not properly expressed unless the direction is included in the answer.

What we determined in the preceding example are the average values for the speed and velocity of the person's travel. If we were to analyze his progress in more detail and take measurements at much shorter time intervals during his trip, we could determine his velocity and speed at specific moments during his hike in the woods. It should be clear that this trip could be made at a constant pace or that the person could run for a while, then stop and admire the scenery, and then go on again. If we consider very short time intervals, we can calculate the instantaneous values for his speed and velocity at specific moments during his trip. The units for velocity and speed both involve length divided by time and so are feet/second, centimeters/second, or as seen in the above example, meters/second.


Section   2.3Acceleration

As mentioned in the last Section, motion does not always occur at a constant rate. Motion that is not at a constant rate involves changes in velocity. The time rate of change in velocity gives us a basic working definition for acceleration. Acceleration is equal to the change in velocity divided by the change in time. If an object is speeding up, we say that it is accelerating; if it is slowing down, it is decelerating. It is possible for the rate of acceleration itself to change as a function of time, but this motion is usually quite complex and is beyond the scope of the mathematical skills required by this course, so only cases of constant, or uniform, acceleration will be considered.

A simple example of acceleration occurs when you are traveling along in your car, say at 35.0 m/s, and must stop for a red light. As you apply the brakes, the car will slow down (decelerate) until it comes to rest. Assuming this occurs in a uniform manner over 7.0 seconds, the acceleration can be calculated as the change in velocity divided by the time interval involved; (0 m/s - 35.0 m/s) / 7.0 s, which gives a value of -5.0 m/s² for the acceleration, or in this case as the negative sign indicates, the deceleration. Notice that the units (m/s)/s have been expressed in the more conventional form of m/s². Can you determine the proper units for acceleration if the velocity had been expressed in feet/second? Did you get ft/s²? That's good, because if you did, you are correct.

An important type of acceleration occurs when any object experiences a fall near Earth's surface. Because of the pull of Earth's gravity, the velocity of the object changes as the object drops, but the time rate of change in the velocity is always very nearly constant at 9.80 m/s² (or 32 ft/s²) as long as air resistance or other friction forces can be neglected. This is quite important because it means that if air friction is very small and can be ignored, all objects in free-fall will behave in a similar way and the final velocities and distances dropped in any specific time interval will all be the same, no matter how large or small the objects are.


Section   2.4Acceleration in Uniform Circular Motion

One concept about motion that is often hard for students to grasp is that an acceleration can occur even when the speed or magnitude of the velocity of an object is constant. This means that the object may not speed up or slow down, but there can still be a change in its velocity. The key here is that the direction of the velocity is changing, even though the magnitude of the velocity is not. Thus any object following a curved path, even when it is traveling at a constant speed, will be experiencing an acceleration. This type of acceleration is referred to as centripetal (center seeking) acceleration because the acceleration vector always points toward the center of the circle whose radius defines the amount of curvature being followed.

Any object that is experiencing a curving motion is traveling along the circumference of a circle. Tight turns have a very short radius, whereas gradual turns have a longer radius. A model airplane fastened to a long string held tightly by a child may follow such a path, as will a bicycle rider going around a turn, or the Moon as it orbits Earth. The value of the centripetal acceleration (a_c) is given by the equation a_c = v ² / rwhere v is the tangential velocity or speed of the object as it moves around the curve and r is the radius of the circular path being followed. Centripetal motion is covered in more detail in the textbook under the discussion of centripetal force in Chapter 3.


Section   2.5Projectile Motion

Because of the continuous pull of gravity on all objects located near Earth's surface, any object moving horizontally through the air will be pulled downward in a curved path. This results in a two-dimensional type of motion that is referred to as projectile motion. To understand projectile motion, it is convenient to consider the horizontal motion (parallel to Earth's surface) and the vertical motion (up and down) as basically independent of each other, with the restriction that the motion and position must be linked by the time element of the flight. The path followed by objects experiencing this type of motion has been found to be a parabola. The textbook contains several diagrams of such motion that you should examine carefully. Note that if an object is not launched horizontally, the initial launch angle has a strong effect on the subsequent motion of any projectile traveling near Earth's surface.

Please don't forget to work on the questions and problems in the textbook, as well as those that have been provided in this study guide. The more practice that you have using the concepts and theories presented in this course, the better you will understand the material and the higher your grades will be on the exams and quizzes that you must take.

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