Projectile

Two particles are launched from the same point on the ground with the same initial speed $v_0$ but at different launch angles. Air resistance is negligible.

A projectile is launched from the ground with an initial speed $v_0$. To be successful, it must pass over a barrier of height $y$ located at a horizontal distance $x$ from the launch point.

As shown in the figure, an object P is located at the top of a flagpole of height $h$. A boy at point O, a horizontal distance $s$ from the base of the flagpole A, shoots a small stone with a slingshot to hit the object P. The initial speed of the stone is $v_0$.

Two particles are launched from the same point on level ground with the same initial speed $v_0$ but at different launch angles, $\alpha_1$ and $\alpha_2$. They move in the same vertical plane and land at the same distance R from the launch point, as shown in Figure (a). Air resistance is negligible.

Two particles are launched from the bottom of a slope inclined at an angle $\theta$ to the horizontal. Both are launched with the same initial speed $v_0$ but at different angles, $\alpha_1$ and $\alpha_2$, measured relative to the inclined plane. Both particles land on the incline at the same distance R up the slope, as shown in Figure (b).

As shown in Figure, a cannon is situated on a hillside plane with an inclination angle of $\theta$. The cannon fires a projectile with an initial velocity of $v_0$ at an elevation angle of $\alpha$ relative to the slope. Air resistance is to be neglected.

An object is launched from the base of a triangular prism resting on a horizontal surface. The left face of the prism has an inclination angle of $\theta$ with the horizontal, and the right face has an inclination angle of $\varphi$. The object is launched with an angle $\alpha$ relative to the horizontal from the bottom corner of the $\theta$-inclined face. The trajectory is such that the object just grazes the peak of the prism and lands at the bottom corner of the $\varphi$-inclined face.

As shown in the figure, two balls, Ball 1 and Ball 2, are launched horizontally from the same point at a height $H$ above the ground. The origin O is directly below the launch point. The initial horizontal velocities are $v_1$ and $v_2$ respectively, with $v_1 > v_2$. Ball 1's trajectory just clears the top of a vertical rod of height $h$ located at a horizontal position $x_P$. Ball 1 then lands on the ground at a horizontal distance $R$ from the origin. Ball 2 is launched, lands on the ground at $x=R/3$, undergoes a perfect elastic collision, and its rebound trajectory also just clears the top of the same rod. Ball 2's second landing point is also at $x=R$.

Two people are playing catch. The ball is thrown and caught at the same height. The horizontal distance between them is $s$, and the total time of flight for the pass is $t$.

As shown in the figure, a water gun needs to shoot water over a wall. The wall is at a horizontal distance of $d=3$ m from the nozzle and has a height of $h=4.0$ m relative to the nozzle. Air resistance is to be neglected. The acceleration due to gravity is $g=10$ m/s².