A rock was thrown from a 225m high place. Below, a person is walking, z(s) = s³ – 15s² + 63s – 49. Let z meters be the distance of the person from the position that is right below the rock at s seconds. a. max height of rock?
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use derivatives rectilinear motion
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- A moving beltway at an airport has a speed 1 and a length L. A woman stands on the beltway as it moves from one end to the other, while a man in a hurry to reach his flight walks on the beltway with a speed of 2 relative to the moving beltway. (a) What time interval is required for the woman to travel the distance L? (b) What time interval is required for the man to travel this distance? (c) A second beltway is located next to the first one. It is identical to the first one but moves in the opposite direction at speed 1. Just as the man steps onto the beginning of the beltway and begins to walk at speed relative to his beltway, a child steps on the other end of the adjacent beltway. The child stands at rest relative to this second beltway. How long after stepping on the beltway does the man pass the child?A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s2, (a) What is the acceleration of the bolt relative to the train car? (b) What is the acceleration of the bolt relative to the Earth? (c) Describe the trajectory of the bolt as seen by an observer fixed on the Earth.A package is dropped from a helicopter that is descending steadily at a speed vn. After t seconds have elapsed, (a) what is the speed of the package in terms of vn, g, and t? (b) What distance d is it from the helicopter in terms of g and t? (c) What are the answers to parts (a) and (b) if the helicopter is rising steadily at the same speed?
- Two students are on a balcony a distance h above the street. One student throws a ball vertically downward at a speed vi, at the same time, the other student throws a ball vertically upward at the same speed. Answer the following symbolically in terms of vi, g, h. and t. (a) What is the time interval between when the first ball strikes the ground and the second ball strikes the ground? (b) Find the velocity of each ball as it strikes the ground, (c) How far apart are the balls at a time t after they are thrown and before they strike the ground?A package is dropped from a helicopter that is descending steadily at a speed vn. After t seconds have elapsed, (a) what is the speed of the package in terms of vn, g, and t? (b) What distance d is it from the helicopter in terms of g and t? (c) What are the answers to parts (a) and (b) if the helicopter is rising steadily at the same speed?Towns A and B in Figure P3.35 are 80.0 km apart. A couple arranges to drive from town A and meet a couple driving from town B at the lake, L. The two couples leave simultaneously and drive for 2.50 h in the directions shown. Car 1 has a speed of 90.0 km/h. If the cars arrive simultaneously at the lake, what is the speed of car 2? Figure P3.35
- Unreasonable Results Suppose you wish to shoot supplies straight up to astronauts in an orbit 36,000 km above the surface of the Earth. (a) At what velocity must the supplies be launched? (b) What is unreasonable about this velocity? (c) Is there a problem with the relative velocity between the supplies and the astronauts when the supplies reach their maximum height? (d) Is the premise unreasonable or is the available equation inapplicable? Explain your answer.A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of 18.0 m/s. The cliff is 50.0 m above a flat, horizontal beach as shown in Figure P3.7. (a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity? (c) Write the equations for the x- and y-components of the velocity of the stone with time, (d) Write the equations for the position of the stone with time, using the coordinates in Figure P3.7. (e) How long after being released does the stone strike the beach below the cliff? (f) With what speed and angle of impact does the stone land? Figure P3.7A police car traveling at 95.0 km/h is traveling west, chasing a motorist traveling at 80.0 km/h. (a) What is the velocity of the motorist relative to the police car? (b) What is the velocity of the police cat relative to the motorist? (c) If they are originally 250 m apart, in what time interval will the police car over take the motorist?
- You drop a ball from a window on an upper floor of a building and ii is caught by a friend on the ground when the ball is moving with speed vf. You now repeat the drop, but you have a friend on the street below throw another ball upward at speed vf exactly at the same time that you drop your ball from the window. The two balls are initially separated by 28.7 m. (a) At what time do they pass each other? (b) At what location do they pass each other relative the window?An athlete crosses a 25-m-wide river by swimming perpendicular to the water current at a speed of 0.5 m/s relative to the water. He reaches the opposite side at a distance 40 m downstream from his starting point. How fast is the water in the river flowing with respect to the ground? What is the speed of the swimmer with respect to a friend at rest on the ground?An ice sled powered by a rocket engine sum from rest on a large frozen lake and accelerates at + 40 ft/s2. After some time t1, the rocket engine is shut down and the sled moves with constant velocity v for a time t2. If the total distance traveled by the sled is 17 500 ft and the total time is 90 s. find (a) the times t1 and t2 and (b) the velocity v. At the 17 500-ft mark, the sled begins to accelerate at 20 ft/s2. (c) What is the final position of the sled when it comes to rest? (d) How long does it take to come to rest?