$A$ particle of mass $m$ moving with velocity $V_0$ strikes a simple pendulum of mass $m$ and sticks to it. The maximum height attained by the pendulum will be

Two objects having masses in the ratio $1:4$ are at rest. When both of them are subjected to the same force separately,they achieve the same kinetic energy during times $t_1$ and $t_2$ respectively. Then the ratio of $t_1$ to $t_2$ is

$A$ small block of mass $M$ moves on a frictionless surface of an inclined plane,as shown in the figure. The angle of the incline suddenly changes from $60^{\circ}$ to $30^{\circ}$ at point $B$. The block is initially at rest at $A$. Assume that collisions between the block and the incline are totally inelastic $\left(g=10 \ m/s^2\right)$.
$1.$ The speed of the block at point $B$ immediately after it strikes the second incline is
$(A) \sqrt{60} \ m/s$ $(B) \sqrt{45} \ m/s$ $(C) \sqrt{30} \ m/s$ $(D) \sqrt{15} \ m/s$
$2.$ The speed of the block at point $C$,immediately before it leaves the second incline is
$(A) \sqrt{120} \ m/s$ $(B) \sqrt{105} \ m/s$ $(C) \sqrt{90} \ m/s$ $(D) \sqrt{75} \ m/s$
$3.$ If the collision between the block and the incline is completely elastic,then the vertical (upward) component of the velocity of the block at point $B$,immediately after it strikes the second incline is
$(A) \sqrt{30} \ m/s$ $(B) \sqrt{15} \ m/s$ $(C) 0$ $(D) -\sqrt{15} \ m/s$
Give the answers for questions $1, 2,$ and $3.$

Two bodies of different masses $m_1$ and $m_2$ have equal momenta. Their kinetic energies $E_1$ and $E_2$ are in the ratio:

$A$ particle of mass $m$ is moving in a horizontal circle of radius $r$ under a centripetal force equal to $-K/r^2$,where $K$ is a constant. The total energy of the particle is

$A$ bob of mass $m$,suspended by a string of length $l_1$,is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point,it collides elastically with another bob of mass $m$ suspended by a string of length $l_2$,which is initially at rest. Both the strings are massless and inextensible. If the second bob,after collision,acquires the minimum speed required to complete a full circle in the vertical plane,the ratio $\frac{l_1}{l_2}$ is