$A$ bullet fired into a fixed target loses half of its velocity after penetrating $1\,cm$. How much further will it penetrate before coming to rest,assuming that it faces constant resistance to motion?

The inclined surfaces of two movable wedges of same mass $M$ are smoothly conjugated with the horizontal plane as shown in the figure. $A$ washer of mass $m$ slides down the left wedge from a height $h$. To what maximum height will the washer rise along the right wedge? Neglect friction.

An engine is attached to a wagon through a shock absorber of length $1.5\, \text{m}$. The system with a total mass of $40,000\, \text{kg}$ is moving with a speed of $72\, \text{km/h}$ when the brakes are applied to bring it to rest. In the process of the system being brought to rest,the spring of the shock absorber gets compressed by $1.0\, \text{m}$. If $90\, \%$ of the energy of the wagon is lost due to friction,the spring constant is $....\, \times 10^{5}\, \text{N/m}$.

$A$ small disc of mass $m$ slides down with initial velocity zero from the top $(A)$ of a smooth hill of height $H$ having a horizontal portion $(BC)$ as shown in the figure. If the height of the horizontal portion of the hill is $h$,then the maximum horizontal distance covered by the disc from the point $D$ is

$A$ body of mass $M$ is dropped from a height $h$ on a sand floor. If the body penetrates $x$ cm into the sand,then the average resistance offered by the sand to the body is

$A$ particle $P$ is sliding down a frictionless hemispherical bowl. It passes the point $A$ at $t = 0$. At this instant of time, the horizontal component of its velocity is $v$. $A$ bead $Q$ of the same mass as $P$ is ejected from $A$ at $t = 0$ along the horizontal string $AB$ (see figure) with the speed $v$. Friction between the bead and the string may be neglected. Let ${t_P}$ and ${t_Q}$ be the respective time taken by $P$ and $Q$ to reach the point $B$. Then