$A$ student of mass $M$ is $1.5 \,m$ tall and has her centre of mass $1 \,m$ above the ground when standing straight. She wants to jump up vertically. To do so,she bends her knees so that her centre of mass is lowered by $0.2 \,m$ and then pushes the ground by a constant force $F$. As a result,she jumps up such that the maximum height of her feet is $0.3 \,m$ above the ground. The ratio $F / Mg$ is (in $.5$)

$A$ ball of mass $0.4 \ kg$ is thrown vertically upwards by applying a force by hand. If the hand moves $0.5 \ m$ while applying the force and the ball goes up to $5 \ m$ height further,find the magnitude of the force (in $N$) $(g = 10 \ m/s^2)$.

$A$ bead $P$ slides on a frictionless semi-circular wire $(ACB)$. It is at point $S$ at $t = 0$, and at this instant, the horizontal component of its velocity is $v$. Another bead $Q$ of the same mass as $P$ is ejected from point $A$ at $t = 0$ along the horizontal wire $AB$ with speed $v$. Friction between the beads and the respective wires may be neglected in both cases. Let $t_P$ and $t_Q$ be the respective times taken by beads $P$ and $Q$ to reach point $B$. Then the relation between $t_P$ and $t_Q$ is:

Explain the work-energy theorem.

$A$ body of mass $m$ starts moving from rest along the $x-$axis such that its velocity varies as $v = a\sqrt{s}$,where $a$ is a constant and $s$ is the distance covered by the body. The total work done by all the forces acting on the body in the first $t$ seconds after the start of the motion is:

$A$ body of mass $1 \ kg$ is under a force, which causes a displacement given by $x = \frac{t^3}{3}$ (in $m$). Find the work done by the force in the first second (in $J$).