Suppose an object could move in three dimensions. What additions to the equations of kinematics in Table 3.1 would be necessary to describe three-dimensional motion?
An object is thrown upward at an angle q above the ground, eventually returning to earth. (a) Is there any place along the trajectory where the velocity and acceleration are perpendicular? If so, where? (b) Is there any place where the velocity and acceleration are parallel? If so, where? In each case, explain.
Is the acceleration of a projectile equal to zero when it reaches the top of its trajectory? If not, why not?
In baseball, the pitcher’s mound is raised to compensate for the fact that the ball falls downward as it travels from the pitcher toward the batter. If baseball were played on the moon, would the pitcher’s mound have to be higher than, lower than, or the same height as it is on earth? Give your reasoning.
A tennis ball is hit upward into the air and moves along an arc. Neglecting air resistance, where along the arc is the speed of the ball (a) a minimum and (b) a maximum? Justify your answers.
Suppose there is a wind blowing parallel to the ground and toward the kicker in Figure 3.12. Then the acceleration component in the horizontal direction would not be zero. How would you expect the time of flight of the football to be affected, if at all? Explain.
Concept Simulation 3.1 provides a review of the concepts that are important in this question. A wrench is accidentally dropped from the top of the mast on a sailboat. Will the wrench hit at the same place on the deck whether the sailboat is at rest or moving with a constant velocity? Justify your answer.
A rifle, at a height H above the ground, fires a bullet parallel to the ground. At the same instant and at the same height, a second bullet is dropped from rest. In the absence of air resistance, which bullet strikes the ground first? Explain.
Two projectiles are launched from ground level at the same angle above the horizontal, and both return to ground level. Projectile A has a launch speed that is twice that of projectile B. Assuming that air resistance is absent, sketch the trajectories of both projectiles. If your drawings are to be accurate, what should be the ratio of the maximum heights in your drawings and what should be the ratio of the ranges? Justify your answers.
A stone is thrown horizontally from the top of a cliff and eventually hits the ground below. A second stone is dropped from rest from the same cliff, falls through the same height, and also hits the ground below. Ignore air resistance. Discuss whether each of the following quantities is different or the same in the two cases; if there is a difference, describe the difference: (a) displacement, (b) speed just before impact with the ground, and (c) time of flight.
A leopard springs upward at a 45° angle and then falls back to the ground. Does the leopard, at any point on its trajectory, ever have a speed that is one-half its initial value? Give your reasoning.
As background for this question, review Conceptual Example 4. A football quarterback throws a pass on the run and then keeps running without changing his velocity. Can he throw the pass and then catch it himself? Give your reasoning.
On a riverboat cruise, a plastic bottle is accidentally dropped overboard. A passenger on the boat estimates that the boat pulls ahead of the bottle by 5 meters each second. Is it possible to conclude that the boat is moving at 5 m/s with respect to the shore? Account for your answer.
A plane takes off at St. Louis, flies straight to Denver, and then returns the same way. The plane flies at the same speed with respect to the ground during the entire flight, and there are no head winds or tail winds. Since the earth revolves around its axis once a day, you might expect that the times for the outbound trip and the return trip differ, depending on whether the plane flies against the earth’s rotation or with it. However, under the conditions given, the two flight times are identical. Explain why.
A child is playing on the floor of a recreational vehicle (RV) as it moves along the highway at a constant velocity. He has a toy cannon, which shoots a marble at a fixed angle and speed with respect to the floor. The cannon can be aimed toward the front or the rear of the RV. Is the range toward the front the same as, less than, or greater than the range toward the rear? Answer this question (a) from the child’s point of view and (b) from the point of view of an observer standing still on the ground. Justify your answers.
Three swimmers can swim equally fast relative to the water. They have a race to see who can swim across a river in the least time. Swimmer A swims perpendicular to the current and lands on the far shore downstream, because the current has swept him in that direction. Swimmer B swims upstream at an angle to the current and lands on the far shore directly opposite the starting point. Swimmer C swims downstream at an angle to the current in an attempt to take advantage of the current. Who crosses the river in the least time? Account for your answer.
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