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 $t_A: t_B=$

$A$ 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

State whether the following statements are true or false:
$(a)$ If the magnitude of force and length are increased by $4$ times,the magnitude of energy increases by $16$ times.
$(b)$ In an inelastic collision,both momentum and energy are conserved.
$(c)$ If work is done on a system by non-conservative forces,the potential energy increases.

It is well known that a raindrop falls under the influence of the downward gravitational force and the opposing resistive force. The latter is known to be proportional to the speed of the drop but is otherwise undetermined. Consider a drop of mass $1.00 \; g$ falling from a height $1.00 \; km$. It hits the ground with a speed of $50.0 \; m s^{-1}$. $(a)$ What is the work done by the gravitational force? $(b)$ What is the work done by the unknown resistive force?

$A$ lorry and a car moving with the same $K.E.$ are brought to rest by applying the same retarding force,then

$A$ 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 $v_1$.