Projectile motion：練習問題

A projectile is launched with a velocity \( v \) at an angle \( \theta \) to the horizontal. Which of the following statements correctly describes the acceleration of the projectile at its maximum height, assuming air resistance is negligible?

A particle is projected with speed \( u \) at an angle \( \alpha \) to the horizontal. At what time \( t \) after projection is the velocity vector perpendicular to the initial velocity vector?

An object is thrown horizontally from the top of a cliff of height \( H \) with an initial speed \( u \). If air resistance is neglected, how does the time taken to reach the ground depend on the initial speed \( u \)?

An airplane flying horizontally at a constant speed of \( 120 \, \text{m s}^{-1} \) at an altitude of \( 500 \, \text{m} \) drops a package. Calculate the magnitude of the velocity of the package just before it hits the ground. Neglect air resistance and use \( g = 9.81 \, \text{m s}^{-2} \).

A ball is projected from horizontal ground with an initial velocity of \( 25 \, \text{m s}^{-1} \) at an angle of \( 35^{\circ} \) to the horizontal. Neglecting air resistance, what is the maximum height reached by the ball? Use \( g = 9.81 \, \text{m s}^{-2} \).

A ball is thrown horizontally with an initial speed of \( 15\text{ m s}^{-1} \) from a height of \( 20\text{ m} \) above level ground. Ignoring air resistance, calculate the magnitude of the velocity of the ball just before it hits the ground. (Take \( g = 9.81\text{ m s}^{-2} \))

Two stones, A and B, are released from the same height. Stone A is dropped vertically from rest, while stone B is thrown horizontally with an initial speed \( v \). By considering the equations of motion for the vertical component, explain why both stones reach the ground at the same time despite their different paths.

A projectile is launched from ground level with an initial velocity \( u \) at an angle \( \theta \) to the horizontal. Show that the horizontal range \( R \) is given by \( R = \frac{u^2 \sin(2\theta)}{g} \) and state the angle for which the range is maximum for a fixed launch speed.

A golf ball is hit with an initial velocity of \( 35 \text{ m s}^{-1} \) at an angle of \( 40^{\circ} \) to the horizontal ground. Air resistance is negligible.
(a) Show that the time taken for the ball to reach its maximum height is approximately \( 2.3 \text{ s} \).
(b) Calculate the horizontal distance (range) traveled by the ball before it hits the ground.
(c) In reality, air resistance affects the motion. Describe and explain the effect of air resistance on the maximum height and the horizontal range of the ball.

A projectile is launched from ground level with an initial speed \( u \) at an angle \( θ \) to the horizontal.
(a) By considering the vertical and horizontal components of the motion independently, derive an expression for the total time of flight \( T \) in terms of \( u, θ, \) and \( g \).
(b) Hence, show that the horizontal range \( R \) is given by \( R = \frac{u^2 \sin(2θ)}{g} \). (Note: \( 2μ \sinθ \cosθ = μ \sin(2θ) \)).
(c) A ball is projected at an angle of \( 30^{\circ} \) and reaches a range of \( 50 \text{ m} \). Calculate the initial speed \( u \).
(d) State the angle \( θ \) that would result in the maximum possible range for a given speed \( u \).