$A$ ball is falling freely from a height. When it reaches $10 ~m$ height from the ground, its velocity is $v_0$. It collides with the ground, loses $50 \%$ of its energy, and rises back to a height of $10 ~m$. Then the velocity $v_0$ is (in $~m / s$)

In the case of circular motion of a body,if a tangential force also acts on the body in addition to a centripetal force,then the work done:

$A$ rubber ball is dropped from a height of $5 \ m$ on a planet,where the acceleration due to gravity is not known. On bouncing,it rises to $1.8 \ m$. The ball loses its velocity on bouncing by a factor of

By how much should the spring be compressed by the block so that the centripetal force at point $P$ is $mg$?

$A$ block of mass $m$ starts at rest at height $h$ on a frictionless inclined plane. The block slides down the plane,travels across a rough horizontal surface with coefficient of kinetic friction $\mu$,and compresses a spring with force constant $k$ a distance $x$ before momentarily coming to rest. Then the spring extends and the block travels back across the rough surface,sliding up the plane. The block travels a total distance $d$ on the rough horizontal surface. The correct expression for the maximum height $h'$ that the block reaches on its return is

Consider a one-dimensional collision involving a body of mass $m_1$ moving in the positive $x$ direction with speed $v_0$ and a second body of mass $m_2$ initially at rest. The collision can be completely inelastic,completely elastic,or partially inelastic. After the collision,$m_1$ moves with velocity $v_1$ and $m_2$ moves with velocity $v_2$. Determine the range of final velocity $v_1$ for the following cases:
$(A)$ If $m_1 > m_2$
$(B)$ If $m_1 < m_2$