$A$ particle of mass $0.01 \; kg$ travels with velocity given by $4 \hat{i} + 16 \hat{k} \; m s^{-1}$. After some time,its velocity becomes $8 \hat{i} + 20 \hat{j} \; m s^{-1}$. The work done on the particle during this interval of time is ...... $J$.

$A$ ball of mass $1 \ kg$ moves in a straight line with velocity $v = c x^\alpha$, where $c = 1$ ($SI$ unit) and $\alpha$ is a constant. If the work done by the net force during its displacement from $x = 0$ to $x = 4 \ m$ is $128 \ J$, then the value of $\alpha$ is

$A$ demonstration apparatus on a table in the lab is shown in the diagram. It consists of a metal track (shown as a thick solid line in the figure) along which a perfectly spherical marble can roll without slipping. In one run,the marble is released from rest at a height $h$ above the table on the left section,rolls down one side and then up the other side without slipping,briefly stopping when it has reached $h_1$. Assuming the table to be horizontal and neglecting air drag as well as any energy loss due to rolling,which of the following is true?

$A$ body thrown vertically upwards from the ground reaches a maximum height $h$. The ratio of the kinetic and potential energies of the body at a height $40 \%$ of $h$ from the ground is

$A$ uniform flexible chain of mass $m$ and length $2l$ hangs in equilibrium over a smooth horizontal pin of negligible diameter. One end of the chain is given a small vertical displacement so that the chain slips over the pin. The speed of the chain when it leaves the pin is

$A$ particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that: