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Chapter Review

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2-1 Position, Distance, and Displacement

Any description of motion takes place in a coordinate system that allows us to track the position of an object. One-dimensional motion means that objects are only free to move back and forth along a single line. As a coordinate system for one-dimensional motion, choose this line to be an x-axis together with a specified origin and positive and negative directions. The location of the origin and which way is called positive or negative may be chosen according to convenience.

The distance traveled by an object that moves from one position to another is just the total length of travel during the trip. This total length of travel depends on the path taken as the object moves from its initial position x_i to its final position x_f. Distance should be distinguished from displacement, = x_f - x_i, which is just the change in position of the object regardless of the path taken. The primary difference between the two is that the distance an object travels tells you nothing about the direction of travel, while displacement tells you precisely how far, and in what direction, from its initial position an object is located. To put it succinctly, distance is the total length of travel and displacement is the net length of travel accounting for direction.

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Example 2.1 Parking in the Same Spot: In your car, you leave your favorite parking spot and drive 4.83 km east on Main Street to go to the grocery store. After shopping, you go back home by traveling west on Main Street and find your favorite parking spot is still available. (a) What distance do you travel during this trip? (b) What is your displacement?

Picture the Problem We choose the x-axis to represent Main Street. The origin is placed at that initial position in the parking lot.

Strategy To solve this problem we must remember the distinction between distance and displacement. For the distance we must consider the entire length of the path taken. For displacement we ignore the path and focus on the initial and final positions only.

(a) The trip has two length segments: s₁ going to the store and s₂ coming back home. The total length of travel, s, must be the sum of these two segments.

(b) Here we notice that the initial and final positions are the same.

Solution

1. (a) Sum the lengths of each segment to get the total length of travel.	s = s1 + s2 = 4.83 km + 4.83 km = 9.66 km
2. (b) Subtract the intial position from the final position to get the displacement.	x = xf - xi = 0 m = 0 m

Insight This example clearly shows how different the distance and displacement can be. It doesn't matter where the origin is placed; the results for both parts (a) and (b) would be the same. As an exercise, you should convince yourself of this fact.

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Practice Quiz

If you walk exactly four times around a quarter-mile track, what is your displacement? one mile one-half mile one-quarter mile zero

2-2 - 2-3 Speed and Velocity

The concepts of distance and displacement, discussed above, relate to the fact that an object moves from one position to another. Additionally, an important part of describing motion is to specify how rapidly an object moves. One way to do this is to quote an average speed,

Thus, average speed is the distance traveled divided by the amount of time it took to travel that distance.

Another, sometimes more appropriate, way to describe the rate of motion is to quote an average velocity,

Thus, average velocity is the displacement divided by the amount of time it took to undergo that displacement. The difference between average speed and average velocity is that average speed relates to the distance traveled while average velocity relates to the displacement. Therefore, average speed tells you about the average rate of motion over the entire path taken and contains no directional information. Average velocity, on the other hand, only relates to the rate at which an object goes from x_i to x_f, regardless of the path taken, and thereby specifies direction.

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Example 2.2 Average Speed versus Average Velocity: For the trip described in example 2.1, if it took t₁ = 10.0 minutes to drive to the store and t₂ = 12.0 minutes to drive back home, calculate the average speed and average velocity for the trip?

Picture the Problem The same picture from example 2.1 applies here.

Strategy Knowing the definitions of average speed and average velocity, we can apply them directly using the results of example 2.1.

Solution

1. To get the answers in SI units let us first determine the elapsed time for the entire trip and convert it to seconds.
2. Convert the distance traveled to meters.	9.66 km = 9.66 x 103 m
3. Use the definition of average speed.
4. Similarly, use the definition of average velocity to calculate its value.

Insight The thing to notice here is that just as distance and displacement can be very different, so can average speed and average velocity. Don't confuse the terms.

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Some situations require more than just the average rate of motion. Often, we require the velocity that an object has at a specific instant in time; this velocity is called the instantaneous velocity. The instantaneous velocity, v, can be defined in terms of the average velocity measured over an infinitesimally small elapsed time,

Thus, the intantaneous velocity would be the velocity that an object has right at t = 2.0 s, for example, instead of the average velocity over a time period t = 2.0 s. The magnitude of the instantaneous velocity of an object (how fast it is going) is its instantaneous speed.

In general, average and instantaneous velocities will have very different values. However, the case of constant velocity motion is special in that the average velocity over any time interval equals the instantaneous velocity at any time. Because of this, the definition of average velocity also serves as an equation that describes constant velocity motion. For this special case, the equation is often written as

x = v t,

where v no longer needs to be called v_av.

Physlet Illustration: Constant Velocity
	Interactive Help on      off
Two balls (one red and one blue) move at constant speeds across the floor, as viewed from above. The distance grid is in cm, and the times are shown in seconds. Can you determine the speed of each ball? Start
Hints Find out how far each ball moves over a specific time interval. Divide the distance traveled by the time it takes to move that distance. Which ball (red or blue) moves faster according to your calculations? Does this result agree with what you appear to see in the animation?

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Example 2.3 Time the Moving Ball: How much time is required for a ball that rolls with a constant velocity of 0.64 m/s to roll across a 2.3 m long table?

Picture the Problem For this problem we will take the length along the tabletop as the x-axis with one end as the origin. The ball is taken to roll in the positive direction.

Strategy From the coordinate system established in the picture and the information given in the problem, it is clear that we know both the displacement and the velocity. Since we can take both x_i and t_i to be zero, we can simply put x = x = 2.3 and t = t.

Solution

1. Solve the constant velocity equation for t.
2. Insert the values to get the numerical solution.

Insight Although this problem may seem short it gets to the core of good problem solving. You should identify the information given to you, as specifically as possible, and relate it to the physics you've been learning. This is especially true for longer problems.

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Exercise 2.4 Playing Catch: A college athlete at softball practice stands 13.2 m away from her friend and throws a ball to him at 26.8 m/s (assumed constant while in the air). If the total time, from when she throws the ball to when she hears the sound of her friend catching it, is 0.531 s, what is the speed at which the sound travels?

Solution Try to sketch a picture of the problem. It has two different parts, the athlete throwing the ball, with constant velocity, to her friend, followed by the sound traveling, with a different constant velocity, from the friend to the softball player. The information supplied by the problem can be listed as:

Given: x = 13.2 m, v_ball = 26.8 m/s, T_tot = 0.531 s; Find: v_sound

We first address the quantity of interest and let it guide us to the next step in the solution. We want to calculate the constant velocity of sound. From the equation for constant velocity we know that v_sound = x/t_sound. Since we already know x we must now find t_sound. What else depends on t_sound? The total time depends on both the time for the ball to travel to the friend and the time for sound to travel back: T_tot = t_ball + t_sound. Therefore, if we can determine t_ball we'll be able to determine t_sound. We can use the constant velocity equation to determine the time for the ball to reach the friend

With this result, we now use the equations involving time to solve for t_sound:

t_sound = T_tot - t_ball = 0.531 s - 0.4925 s = 0.0385 s

Finally, use the constant velocity equation to solve for v_sound:

There are several things to notice about this problem. At first glance it may not appear as a problem that can be solved using the constant velocity equation since there is clearly more than one velocity involved. However, by dividing the problem into two segments we were able to apply constant velocity motion to each. Also be sure you understand why the same value of x was used for both the ball and the sound. Here x actually represents distance, the distance between the athlete and her friend, the displacements of the ball and the sound in this problem are of opposite signs. Finally, you'll notice that my answer for t_ball contains four digits instead of three. This is because itis only an intermediate step. As discussed in chapter 1 of this study guide, you should retain an extra digit in intermediate calculations to help avoid excessive round off error.

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Practice Quiz

Which of the following is the correct SI unit for the product of velocity and time? m m/s s s/m m/s2

In a coordinate system in which east is the positive direction and west is the negative direction, you take a total time of 105 seconds to walk 20 m west, then 10 m east, followed by 15 m west. With what average speed have you walked? 0.24 m/s 0.43 m/s -0.24 m/s -0.43 m/s 0 m/s

For the information given in question #3 above, with what average velocity have you walked? 0.24 m/s 0.43 m/s -0.24 m/s -0.43 m/s 0 m/s

How long does it take a person on a bicycle to travel exactly one kilometer if she rides at a constant velocity of 20 m/s? 20000 s 0.020 s 50 s 20 s 5.0 s

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