Relative Motion

A truck's canopy covers the cargo area only up to point $A$; a passenger may sit at the rear point $B$. The line $AB$ makes an angle $\varphi = 30°$ with the vertical, and point $C$ lies on the truck bed directly below $A$. The truck drives through rain on a level straight road. When its speed is $u_1 = 6$ km/h, point $C$ is just barely not hit by raindrops; when its speed is $u_2 = 18$ km/h, point $B$ is just barely not hit.

Rain falls so that each raindrop moves uniformly in a straight line relative to the ground, falling vertically with speed $v$. A person rides in a car that moves in the $x$ direction with constant acceleration $a$. Observation starts at $t = 0$, when the car has speed $u_0$ and a raindrop is at height $y_0$ on the $y$-axis (directly above the car's position).

In still air an airplane flies with uniform speed $v$ relative to the ground, and the farthest distance it can fly out (counting the trip out and back) is $R$. The airplane now flies in a wind of speed $u$ blowing toward the direction $\alpha$ degrees east of north, while the airplane's actual track over the ground is $\beta$ degrees east of north (out and back along the same line).

A ship sails parallel to a straight coastline at distance $D$ from the shore with speed $V$. A speedboat with speed $v$ ($v < V$) starts from a port on the coastline to intercept the ship.

Particles $A$ and $B$ start simultaneously from points $A$ and $B$, moving in uniform rectilinear motion with speed $v_1$ along $AB$ and speed $v_2$ along $BC$ respectively, where the angle between segments $AB$ and $BC$ (the angle $\angle ABC$) is $\alpha$. Initially the particles are a distance $l$ apart.

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)