Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 81 | Concept Question Listed below are three pairs of initial and final positions that lie along the x axis. Using the concept of displacement discussed in Section 2.1, decide which pairs give a positive displacement and which a negative displacement. Explain.

| Initial position x0 | Final position x
(a) | +2.0 m | +6.0 m
(b) | +6.0 m | +2.0 m
(c) | −3.0 m | +7.0 m

Problem For each of the three pairs of positions listed above, determine the magnitude and direction (positive or negative) of the displacement. Verify that your answers are consistent with those determined in the Concept Question.

| 82 | Concept Question Suppose that each pair of positions in Problem 81 represents the initial and final positions of a moving car. (a) In each case, is the direction of the car’s average velocity the same as its initial position x0, its final position x, or the difference x − x0 between its final and initial positions? (b) Decide which pairs give a positive average velocity and which give a negative average velocity. Provide reasons for your answers.
Problem The elapsed time for each pair of positions is 0.50 s. Review the concept of average velocity in Section 2.2 and then determine the average velocity (magnitude and direction) for each of the three pairs. Check to see that the directions of the velocities agree with the directions found in the Concept Question.

| 83 | Concept Questions (a) In general, does the average acceleration of an object have the same direction as its initial velocity v0, its final velocity v, or the difference v − v0 between its final and initial velocities? Provide a reason for your answer. (b) The following table lists four pairs of initial and final velocities for a boat traveling along the x axis. Use the concept of acceleration presented in Section 2.3 to determine the direction (positive or negative) of the average acceleration for each pair of velocities.

| Initial velocity v0 | Final velocity v
(a) | +2.0 m/s | +5.0 m/s
(b) | +5.0 m/s | +2.0 m/s
(c) | −6.0 m/s | −3.0 m/s
(d) | +4.0 m/s | −4.0 m/s

Problem The elapsed time for each of the four pairs of velocities is 2.0 s. Find the average acceleration (magnitude and direction) for each of the four pairs. Be sure that your directions agree with those found in the Concept Questions.

| 84 | Concept Questions The initial velocity v0 and acceleration a of four moving objects at a given instant in time are given in the table:

| Initial velocity v0 | Acceleration a
(a) | +12 m/s | +3.0 m/s2
(b) | +12 m/s | −3.0 m/s2
(c) | −12 m/s | +3.0 m/s2
(d) | −12 m/s | −3.0 m/s2

Draw vectors for v0 and a and, in each case, state whether the speed of the object is increasing or decreasing in time. Account for your answers.
Problem For each of the four pairs in the table above, determine the final speed of the object if the elapsed time is 2.0 s. Compare your final speeds with the initial speeds and make sure that your answers are consistent with your answers to the Concept Questions.

| 85 | Concept Questions (a) The acceleration of a NASCAR race car is zero. Does this necessarily mean that the velocity of the car is also zero? (b) If the speed of the car is constant as it goes around the track, is its average acceleration necessarily zero? Justify your answers.
Problem (a) Suppose that the race car is moving to the right with a constant velocity of +82 m/s. What is the average acceleration of the car? (b) Twelve seconds later, the car is halfway around the track and traveling in the opposite direction with the same speed. What is the average acceleration of the car? Verify that your answers agree with your answers to the Concept Questions.

| 86 | Concept Questions Concept Simulation 2.3 at provides some background for this problem. A ball is thrown vertically upward, which is the positive direction. A little later it returns to its point of release. (a) Does the acceleration of the ball reverse direction when the ball starts its downward trip? (b) What is the displacement of the ball when it returns to its point of release? Explain your answers.
Problem If the ball is in the air for a total time of 8.0 s, what is its initial velocity?

| 87 | Concept Questions Two cars cover the same distance in a straight line. Car A covers the distance at a constant velocity. Car B starts from rest, maintains a constant acceleration, and covers the distance in the same time as car A. (a) Is car B’s final velocity smaller than, equal to, or greater than car A’s constant velocity? Explain. (b) Is car B’s average velocity smaller than, equal to, or greater than car A’s constant velocity? Why? (c) Given a value for the time to cover the distance and the fact that car B starts from rest, what other variable must be known before car B’s acceleration can be determined? Provide a reason for your answer.
Problem Both cars cover a distance of 460 m in 210 s. Assume that they are moving in the +x direction. Determine (a) the constant velocity of car A, (b) the final velocity of car B, and (c) the acceleration of car B. Verify that your answers are consistent with your answers to the Concept Questions.

| 88 | Concept Questions Two stones are thrown simultaneously, one straight upward from the base of a cliff and the other straight downward from the top of the cliff. The stones are thrown with the same speed. (a) Does the stone thrown upward gain or lose speed as it moves upward? Why? (b) Does the stone thrown downward gain or lose speed as time passes? Explain. (c) The speed at which the stones are thrown is such that they cross paths. Where do they cross paths, above, at, or below the point that corresponds to half the height of the cliff? Justify your answer.
Problem The height of the cliff is 6.00 m, and the speed with which the stones are thrown is 9.00 m/s. Find the location of the crossing point. Check to see that your answer is consistent with your answers to the Concept Questions.

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