A projectile is fired from the top of a cliff of height h above the ocean below. The projectile is fired at an angle \(\theta\) above the horizontal and with an initial speed \(v_i\). (a) Find a symbolic expression in terms of the variables \(v_i\), g, and \(\theta\) for the time at which the projectile reaches its maximum height. (b) Using the result of part (a), find an expression for the maximum height \(h_{max}\) above the ocean attained by the projectile in terms of h, \(v_i\), g, and \(\theta\).

In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is h. The mug slides off the counter and strikes the floor at distance d from the base of the counter. (a) With what velocity did the mug leave the counter? (b) What was the direction of the mug’s velocity just before it hit the floor?

Mayan kings and many school sports teams are named for the puma, cougar, or mountain lion—Felis concolor— the best jumper among animals. It can jump to a height of 12.0 ft when leaving the ground at an angle of \(45.0^{\circ}\). With what speed, in SI units, does it leave the ground to make this leap?

A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. What is the angle of projection?

The speed of a projectile when it reaches its maximum height is one-half its speed when it is at half its maximum height. What is the initial projection angle of the projectile?

A rock is thrown upward from level ground in such a way that the maximum height of its flight is equal to its horizontal range R. (a) At what angle \(\theta\) is the rock thrown? (b) In terms of its original range R, what is the range \(R_{max}\) the rock can attain if it is launched at the same speed but at the optimal angle for maximum range? (c) What If? Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet? Explain.

A firefighter, a distance d from a burning building, directs a stream of water from a fire hose at angle \(\theta_i\) above the horizontal as shown in Figure P4.11. If the initial speed of the stream is \(v_i\), at what height h does the water strike the building?

A basketball star covers 2.80 m horizontally in a jump to dunk the ball (Fig. P4.12a). His motion through space can be modeled precisely as that of a particle at his center of mass, which we will define in Chapter 9. His center of mass is at elevation 1.02 m when he leaves the floor. It reaches a maximum height of 1.85 m above the floor and is at elevation 0.900 m when he touches down again. Determine (a) his time of flight (his “hang time”), (b) his horizontal and (c) vertical velocity components at the instant of takeoff, and (d) his takeoff angle. (e) For comparison, determine the hang time of a whitetail deer making a jump (Fig. P4.12b) with center-of-mass elevations \(y_i\) = 1.20 m, \(y_{max}\) = 2.50 m, and \(y_f\) = 0.700 m.

A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of \(v_i\) = 18.0 m/s. The cliff is h 5 50.0 m above a body of water as shown in Figure P4.13. (a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity of the stone? (c) What is the appropriate analysis model for the vertical motion of the stone? (d) What is the appropriate analysis model for the horizontal motion of the stone? (e) Write symbolic equations for the x and y components of the velocity of the stone as a function of time. (f) Write symbolic equations for the position of the stone as a function of time. (g) How long after being released does the stone strike the water below the cliff? (h) With what speed and angle of impact does the stone land?

The record distance in the sport of throwing cowpats is 81.1 m. This record toss was set by Steve Urner of the United States in 1981. Assuming the initial launch angle was \(45^{\circ}\) and neglecting air resistance, determine (a) the initial speed of the projectile and (b) the total time interval the projectile was in flight. (c) How would the answers change if the range were the same but the launch angle were greater than \(45^{\circ}\)? Explain.

A home run is hit in such a way that the baseball just clears a wall 21.0 m high, located 130 m from home plate. The ball is hit at an angle of \(35.0^{\circ}\) to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall, and (c) the velocity components and the speed of the ball when it reaches the wall. (Assume the ball is hit at a height of 1.00 m above the ground.)

A projectile is fired from the top of a cliff of height h above the ocean below. The projectile is fired at an angle \(\theta\) above the horizontal and with an initial speed \(v_i\). (a) Find a symbolic expression in terms of the variables \(v_i\), g, and \(\theta\) for the time at which the projectile reaches its maximum height. (b) Using the result of part (a), find an expression for the maximum height \(h_{max}\) above the ocean attained by the projectile in terms of h, \(v_i\), g, and \(\theta\).

A boy stands on a diving board and tosses a stone into a swimming pool. The stone is thrown from a height of 2.50 m above the water surface with a velocity of 4.00 m/s at an angle of \(60.0^{\circ}\) above the horizontal. As the stone strikes the water surface, it immediately slows down to exactly half the speed it had when it struck the water and maintains that speed while in the water. After the stone enters the water, it moves in a straight line in the direction of the velocity it had when it struck the water. If the pool is 3.00 m deep, how much time elapses between when the stone is thrown and when it strikes the bottom of the pool?