拋體運動：練習題

A projectile is launched from ground level with an initial velocity of \(20\text{ m s}^{-1}\) at an angle of \(30^\circ\) to the horizontal. Assuming the ground is perfectly flat and air resistance is negligible, what is the horizontal range of the projectile? (Take \(g = 9.81\text{ m s}^{-2}\))

A ball is projected horizontally from the top of a cliff of height \(h\) with a velocity \(u\). It lands at a horizontal distance \(d = 2h\) from the base of the cliff. What is the ratio of the ball's initial kinetic energy to its kinetic energy just before it hits the ground? (Ignore air resistance).

A ball is projected horizontally from the top of a tower with an initial speed of \( 12.0\, \text{m s}^{-1} \). It hits the level ground below exactly \( 3.00\, \text{s} \) later. At what angle to the horizontal does the ball strike the ground?
Take \( g = 9.81\, \text{m s}^{-2} \) and ignore air resistance.

State the value of the horizontal acceleration for a projectile moving through a uniform gravitational field and explain why this value occurs when air resistance is considered negligible.

A ball is projected horizontally from a cliff of height \( h \). Explain the effect on the time of flight and the horizontal range if the initial horizontal velocity is doubled, assuming air resistance is negligible.

A stone is projected horizontally from the top of a vertical cliff that is \(20.0\text{ m}\) high. If the stone strikes the ground at an angle of \(45.0^\circ\) to the horizontal, calculate its initial horizontal projection speed. Assume air resistance is negligible and use \(g = 9.81\text{ m s}^{-2}\).

A golf ball is struck from ground level with an initial velocity of \(32.0\text{ m s}^{-1}\) at an angle of \(30.0^{\circ}\) to the horizontal. Air resistance is negligible. Take \(g = 9.81\text{ m s}^{-2}\).

(a) Calculate the vertical and horizontal components of the initial velocity.
(b) Calculate the time taken for the ball to reach its maximum height.
(c) Determine the maximum height reached by the ball.
(d) Calculate the horizontal range of the ball when it returns to the ground.

A small projectile is launched from ground level with an initial speed of \(25.0\text{ m s}^{-1}\) at an angle of \(40.0^\circ\) to the horizontal. A long horizontal ledge is located at a vertical height of \(5.00\text{ m}\) above the level ground. Air resistance is negligible. Take \(g = 9.81\text{ m s}^{-2}\).

(a) Show that there are two distinct times during the flight when the projectile is at the same height as the ledge, and calculate these two values of time.
(b) Calculate the horizontal distance between the two points in space where the projectile passes the height of the ledge.
(c) Determine the magnitude of the velocity (speed) and the angle of the velocity vector relative to the horizontal at the instant the projectile strikes the ledge while it is descending.