$A$ 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 ball starts moving up the inclined face of the wedge and rises to a height $h$. Suppose the particle,when it reaches the horizontal surface again,has a velocity $v_1$ with respect to the ground,and the wedge has a velocity $v_2$. Choose the correct statement$(s)$.

$A$ particle is moving in a circular path of radius $a$ under the action of an attractive potential $U = - \frac{k}{2r^2}$. Its total energy is

$A$ $2\,kg$ block slides on a horizontal floor with a speed of $4\,m/s$. It strikes an uncompressed spring and compresses it until the block is motionless. The kinetic friction force is $110\,N$ and the spring constant is $1000\,N/m$. The spring compresses by ........ $cm$.

$A$ body of mass $m$ is dropped from a height of $h$. Simultaneously,another body of mass $2m$ is thrown vertically upward with such a velocity $v$ that they collide at a height $h/2$. If the collision is perfectly inelastic,the velocity at the time of collision with the ground will be:

Statement $(I)$: The slope of the kinetic energy-displacement curve of a body in motion is directly proportional to its acceleration.
Statement $(II)$: From a height of $15 \ m$, a ball is projected vertically upwards with a velocity of $30 \ m/s$. If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is $30 \%$.
Statement $(III)$: The velocity acquired by a body of mass '$m$' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass '$m$'.
Which of the following is correct?

$A$ $2 \ kg$ block slides on a horizontal surface at a speed of $4 \ m/s$ and strikes an uncompressed spring. The kinetic friction force is $15 \ N$ and the spring constant is $10,000 \ N/m$. By how many $cm$ will the spring be compressed?