A small ball thrown at an initial velocity $v_0$ at an angle $\alpha$ to the horizontal strikes a vertical wall moving towards it at a horizontal velocity $v$ and is bounced to the point from which it was thrown. Determine the time $t$ from the beginning of motion to the moment of impact, neglecting friction losses.

A small ball moves at a constant velocity $v$ along a horizontal surface and at point $\lambda$ falls into a vertical well of depth $H$ and radius $r$. The velocity $v$. of the ball forms an angle $\alpha$ with the diameter of the well drawn through point $A$ (Fig.).
Determine the relation between $v,H,r$, and $\alpha$ for which the ball can "get out" of the well after elastic impacts with the walls. Friction losses should be neglected. (Here, $g$ denotes acceleration due to gravity)

A cannon fires from under a shelter inclined at an angle $\alpha$ to the horizontal (Fig.). The cannon is at point $A$ at a distance $l$ from the base of the shelter (point $B$). The initial velocity of the shell is $v_0$, and its trajectory lies in the plane of the figure.

Determine the maximum range $L_{\max }$ of the shell.

The slopes of the windscreen of two motorcars are ${\beta }_1=30°$ and ${\beta }_2=15°$ respectively. At what ratio $\frac{v_1}{v_2}$ of the velocities of the cars will their drivers see the hailstones bounced by the windscreen of their cars in the vertical direction? Assume that hailstones fall vertically.

A sheet of plywood moves over a smooth horizontal surface. The velocities of points $A$ and $B$ are equal to $v$ and lie in the plane of the sheet (Fig.).
Determine the velocity of point $C$.

A car must be parked in a small gap between the cars parked in a row along the pavement. Should the car be driven out forwards or backwards for the manoeuvre if only its front wheels can be turned?

An aeroplane flying along the horizontal at a velocity $v_0$ starts to ascend, describing a circle in the vertical plane. The velocity of the plane changes with height $h$ above the initial level of motion according to the law $v^2=v_0^2-2a_{0}h$. The velocity of the plane at the upper point of the trajectory is $v_1=\frac{v_0}{2}$.

Determine the acceleration $a$ of the plane at the moment when its velocity is directed vertically upwards.

An open merry-go-round rotates at an angular velocity $\omega$. A person stands in it at a distance $r$ from the rotational axis. It is raining, and the raindrops fall vertically at a velocity $v_0$.

How should the person hold an umbrella to protect himself from the rain in the best way?

A bobbin rolls without slipping over a horizontal surface so that the velocity $v$ of the end of the thread (point $A$) is directed along the horizontal. A board hinged at point $B$ leans against the bobbin (Fig.). The inner and outer radii of the bobbin are $r$ and $R$ respectively.

Determine the angular velocity $\omega$ of the board as a function of an angle $\alpha$.