$A$ particle of mass $m$ moving horizontally with velocity $v_0$ strikes a smooth wedge of mass $M$,as shown in the figure. After the collision,the particle starts moving up the inclined face of the wedge and rises to a height $h$. The final velocity of the wedge $v_2$ is

The block starts from rest as shown in the figure. Find the work done by the force of $10 \,N$ and friction in the time $0$ to $4 \,s$. [Take,$g=10 \,m/s^2$]

$A$ 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

$A$ box of mass $1 \ kg$ is placed on a horizontal surface of length $1 \ m$. $A$ force of $8 \ N$ is applied to move it horizontally by $1 \ m$,while simultaneously,its vertical height is increased by $2 \ m$. What is the total work done in Joules? (Take $g = 10 \ m/s^2$)

As shown in the figure,a block of mass $2 \ kg$ is released from point $P$. It slides $0.5 \ m$ on the inclined plane before hitting a spring. The spring constant is $4000 \ N/m$. The coefficient of friction between the block and the inclined plane is $0.3$. The compression in the spring will be ............... $mm$.

$A$ force exerts an impulse $I$ on a particle changing its speed from $u$ to $2u$. The applied force and the initial velocity are oppositely directed along the same line. The work done by the force is