拋體運動：練習題

A projectile is fired from horizontal ground with an initial speed of \(40\, \text{m s}^{-1}\) at an angle of \(30^\circ\) above the horizontal. Assuming \(g = 10\, \text{m s}^{-2}\), what is the horizontal range of the projectile?

The Cartesian equation of the trajectory of a particle projected from a point \(O\) on a horizontal plane is given by \(y = \sqrt{3}x - \frac{gx^2}{40}\), where \(x\) and \(y\) are horizontal and vertical displacements from \(O\) respectively. Determine the initial speed \(V\) of the particle.

A projectile is launched with speed \(u\) at an angle \(\alpha\). It is found that the speed of the particle at its maximum height is exactly \(\sqrt{\frac{2}{3}}\) times its speed when it is at half its maximum height. What is the value of \(\alpha\)?

A projectile is launched from ground level with an initial speed \(u\). When the particle reaches a height of \(10\, \text{m}\), its speed is observed to be \(20\, \text{m s}^{-1}\). Calculate the initial speed \(u\). (Take \(g = 10\, \text{m s}^{-2}\))

A projectile is fired with an initial speed \(u\) at an angle \(\alpha > 45^\circ\) to the horizontal. At what time \(t\) after projection is the velocity vector of the projectile perpendicular to its initial velocity vector?

A stone is projected from a point O on horizontal ground with speed $20 \text{ m s}^{-1}$ at an angle of elevation $\alpha$, where $\tan \alpha = 4/3$. Assuming the acceleration due to gravity is $g = 10 \text{ m s}^{-2}$.

(a) Determine the initial horizontal and vertical components of the velocity.

(b) Calculate the time taken for the stone to reach its maximum height.

(c) Calculate the greatest height reached by the stone above the ground.

A projectile is fired from the origin O with speed $U$ at an angle $\alpha$ to the horizontal.

(a) Derive the Cartesian equation of the trajectory in the form $y = x \tan \alpha - \frac{g x^2}{2 U^2 \cos^2 \alpha}$.

(b) Given that the projectile passes through the point $(40, 10)$ and the initial angle of projection is $\alpha = 30^\circ$. Using $g = 10 \text{ m s}^{-2}$, find the initial speed $U$ in $\text{m s}^{-1}$.

(c) Calculate the time of flight required for the projectile to travel from O to the point $(40, 10)$.