$A$ body is projected vertically upwards with an initial velocity of $7 \, m/s$. At what height will its kinetic energy become half of its initial value (in $, m$)?

$A$ force $\overrightarrow{F} = 4 \hat{i} - 15 \hat{j} \text{ N}$ acts on a body, resulting in a displacement of $\overrightarrow{S} = 6 \hat{i} \text{ m}$. If the body had a kinetic energy of $7 \text{ J}$ at the beginning of the displacement, what is the kinetic energy at the end of the displacement (in $\text{ J}$)?

State the principle of conservation of mechanical energy for a conservative force.

$A$ small block slides down from rest at point $A$ on the surface of a smooth cylinder,as shown. At point $B$,the block falls off (leaves) the cylinder. The equation relating the angles $\theta_1$ and $\theta_2$ is given by

$A$ uniform chain of mass $M$ and length $L$ is released from rest from a smooth horizontal surface as shown in the figure. The velocity of the chain at the instant it completely comes off the table will be:

$A$ bead of mass $\frac{1}{2} \, kg$ starts from rest from point $A$ to move in a vertical plane along a smooth fixed quarter ring of radius $5 \, m$,under the action of a constant horizontal force $F = 5 \, N$ as shown. The speed of the bead as it reaches point $B$ is ................ $m/s$. [Take $g = 10 \, m/s^2$]