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 $\alpha$.

A magnetic tape is wound on an empty spool rotating at a constant angular velocity. The final radius $r_2$ of the winding was found to be three times as large as the initial radius $r_1$ (Figure). The winding time of the tape is $t_1$.

What is the time $t_2$ required for winding a tape whose thickness is half that of the initial tape?

Two rings $O$ and $O^{\text{'}}$are put on two vertical stationary rods $AB$ and $A^{\text{'}}{B}^{\text{'}}$ respectively. An inextensible thread is fixed at point $A^{\text{'}}$ and on ring $O$ and is passed through ring $O^{\text{'}}$ (Fig.).

Assuming that ring $O^{\text{'}}$ moves downwards at a constant velocity $v_1$, determine the velocity $v_2$ of ring $O$ if $\angle AO{O}^{\text{'}}=\alpha$

.

A weightless inextensible rope rests on a stationary wedge forming an angle $\alpha$ with the horizontal (Fig.). One end of the rope is fixed to the wall at point $A$. A small load is attached to the rope at point $B$. The wedge starts moving to the right with a constant acceleration $a$.

Determine the acceleration $a_1$ of the load when it is still on the wedge.