$A$ small block of mass $m$ is placed on $a$ wedge of mass $M$ as shown, which is initially at rest. All the surfaces are frictionless . The spring attached to the other end of wedge has force constant $k$. If $a'$ is the acceleration of $m$ relative to the wedge as it starts coming down and $A$ is the acceleration acquired by the wedge as the block starts coming down, then  Maximum retardation of $M$ is:

A ring of mass $m$ is attached to a horizontal spring of spring constant $k$ and natural length $l_0$ . Other end of spring is fixed and ring can slide on a smooth horizontal rod as shown. Now the ring is shifted to position $B$ and released, speed of ring when spring attains it's natural length is

$10\  m$ is the total mass of a cannon that includs  all shell. Initial cannon is moving with velocity $10\  m$ is along a horizontal frictionless path. If cannon fires $'n$' shells of mass $m$ in the direction of motion of the cannon one by one with velocity $u$ with respect to ground. (neglect any friction force)

Two particles with mass $m_1$ = $16\ kg$ and $m_2$ = $2\ kg$ slide as unit with a common velocity of $12\ ms^{-1}$ on a level frictionless surface. Between them is a compressed massless spring with spring constant $k$ = $100\ Nm^{-1}$ . The spring, originally compressed by $25\ cm$ , is suddenly released, sending the two masses, which are connected to the spring, flying apart from each other. The orientation of the spring w.r.t. the initial velocity is shown in diagram. What is the relative velocity of separation in $ms^{-1}$ , after the particles lose contact? ................$m/s$

A mass of $0.5\,kg$ moving with a speed of $1.5 \,m/s$ on a horizontal smooth surface, collides with a nearly weightless spring of force constant $k = 50\;N/m$. The maximum compression of the spring would be ............. $\mathrm{m}$

If a long spring is stretched by $0.02\, m$, its potential energy is $U$. If the spring is stretched by $0.1\, m$ then its potential energy will be