$A$ man throws a ball to a height of $12 \ m$,where it reaches with a speed of $12 \ m/s$. If the man throws the ball such that it just reaches this height,what percentage of energy would he have saved?

$A$ body of mass $m_1$ moving with an unknown velocity $v_1 \hat{i}$ undergoes a one-dimensional collision with a body of mass $m_2$ moving with velocity $v_2 \hat{i}$. After the collision,the bodies $m_1$ and $m_2$ move with velocities $v_3 \hat{i}$ and $v_4 \hat{i}$ respectively. If $m_2 = 0.5\, m_1$ and $v_3 = 0.5\, v_1$,find $v_1$.

$A$ bullet of mass $10\, g$ moving horizontally with a velocity of $400\, m s^{-1}$ strikes a wood block of mass $2\, kg$ which is suspended by a light inextensible string of length $5\, m$. As a result,the centre of gravity of the block is found to rise a vertical distance of $10\, cm$. The speed of the bullet after it emerges out horizontally from the block will be ................... $m s^{-1}$.

Two bodies $A$ and $B$ of masses $20 \ kg$ and $5 \ kg$ respectively are at rest. Due to the action of a force of $40 \ N$ separately,if the two bodies acquire equal kinetic energies in times $t_A$ and $t_B$ respectively,then $t_A: t_B=$

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

$A$ baseball having mass of $0.4 \ kg$ is thrown such that one of the forces acting on it varies with time as shown in the first graph. Also,the velocity of the ball is in the same direction as the force. The velocity varies with time as shown in the second graph. Choose the incorrect option (up to $0.3 \ s$).