$A$ toy of mass $20 \,g$ at rest acquires a velocity $(3 \hat{i}-2 \hat{j}) \,m/s$ in $2 \,s$. The power of the toy is: (in $\,W$)

$A$ bead of mass $m$ slides without friction on the wall of a vertical circular hoop of radius $R$ as shown in the figure. The bead moves under the combined action of gravity and a massless spring of constant $k$ attached to the bottom of the hoop. The natural (equilibrium) length of the spring is $R$. If the bead is released from the top of the hoop with negligible initial speed,what is the velocity of the bead when the length of the spring becomes $R$? ($g$ is the acceleration due to gravity)

$A$ raindrop of mass $1.00 \, g$ falling from a height of $1 \, km$ hits the ground with a speed of $50 \, m s^{-1}$. Calculate
$(a)$ the loss of $PE$ of the drop
$(b)$ the gain in $KE$ of the drop
$(c)$ Is the gain in $KE$ equal to loss of $PE$? If not,why?
Take $g = 10 \, m s^{-2}$.

$A$ mass of $10 \, g$ moving with a velocity of $100 \, cm/s$ strikes a pendulum bob of mass $10 \, g$. The two masses stick together. The maximum height reached by the system now is ............... $cm$ $(g = 10 \, m/s^2)$.

Four particles $A, B, C$ and $D$ of equal mass $m$ are placed at four corners of a square. They move with equal uniform speed $v$ towards the intersection of the diagonals. After collision,$A$ comes to rest,$B$ traces its path back with the same speed $v$,and $C$ and $D$ move with equal speeds $v'$. What is the velocity of $C$ after collision?

Two particles,of masses $M$ and $2M$,moving as shown,with speeds of $10\, m/s$ and $5\, m/s$,collide elastically at the origin. After the collision,they move along the indicated directions with speeds $v_1$ and $v_2$ respectively. The values of $v_1$ and $v_2$ are nearly: