Projectile

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.

A small ball is launched horizontally from the top of a building with an initial speed of $v_0 = 10$ m/s. Air resistance is negligible. At a certain point in its trajectory, the magnitude of the ball's normal acceleration is $a_n = 5$ m/s².

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).

From a fixed point A at a height $h$ above the ground, ball Jia is launched with speed $v_0$ and angle $\alpha$ ($0 < \alpha < \pi/2$). It undergoes a perfectly elastic collision with a very massive plate OG, tilted at angle $\theta$ ($0 < \theta < \pi/2$). The collision point is at the same height as A, and the ball returns exactly to A. Simultaneously, another ball, Yi, is dropped from rest from point A and undergoes a perfectly elastic collision with the ground. Consider two possible scenarios where the ball can return to point A.

Several frictionless inclined planes share a common base but have different inclination angles. An object starts from rest at the top of each incline and slides freely down.

A boat moves along a river in uniform rectilinear motion with speed $v$. A person on the boat throws an object obliquely forward (in the direction of the boat's motion) with speed $v_0$ relative to the boat, where $v_0$ makes an angle $\theta$ with the horizontal.

A particle moves in the $Oxy$ plane with motion equations $x = 3t$, $y = 8 - t^2$, where $x$ and $y$ are in meters and $t$ is in seconds.

An object is thrown with initial speed $v_0 = 20$ m/s at an elevation angle $\alpha = 60°$. Air resistance is neglected; take $g = 9.8$ m/s$^2$.

Several small balls are thrown simultaneously from the same point in air, in all directions, each with the same speed $v_0$. Air resistance is neglected.

Two horizontal tracks lie in the same vertical plane, separated by a height $h$. Objects $A$ and $B$, one on each track, are connected by an inextensible light rope passing over a fixed pulley $O$ located at the level of the lower track. Object $A$ moves along the lower track with uniform speed $v$ (away from the pulley), while $B$ slides along the upper track toward the point above $O$. At the instant when the rope segment between the tracks makes a 30° angle with the tracks, a small water droplet $P$ sitting at the midpoint of segment $OB$ (at rest relative to the rope) detaches from the rope. The rope length $BO$ is much larger than the pulley diameter. (15th National High School Physics Competition, second round)