$A$ 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 ...... $J$.

$A$ 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?

$A$ bullet fired from a gun into a large wooden block loses half of its velocity after traveling $6 \ cm$ inside the block. How much further distance (in $cm$) will it travel before coming to rest?

$A$ particle of mass $m$ slides down along a frictionless inclined plane $AC$ as shown in the figure. The kinetic energy of the particle at point $C$ is:

$A$ balloon filled with helium rises against gravity,increasing its potential energy. The speed of the balloon also increases as it rises. How do you reconcile this with the law of conservation of mechanical energy? You can neglect viscous drag of air and assume that the density of air is constant.

$A$ vertical spring with force constant $k$ is fixed on a table. $A$ ball of mass $m$ at a height $h$ above the free upper end of the spring falls vertically on the spring so that the spring is compressed by a distance $d$. The net work done in the process is