$A$ force acts on a $3\, g$ particle in such a way that the position of the particle as a function of time is given by $x = 3t - 4t^2 + t^3$,where $x$ is in $meters$ and $t$ is in $seconds$. The work done during the first $4\, seconds$ is ................. $mJ$.

Three particles $A, B$, and $C$ are thrown from the top of a tower with the same speed $u$. Particle $A$ is thrown upwards, $B$ is thrown downwards, and $C$ is thrown horizontally. If they hit the ground with speeds $V_A, V_B$, and $V_C$ respectively, then:

$A$ small sphere of mass $m$ and radius $r$ slides down the smooth surface of a large hemispherical bowl of radius $R$. If the sphere starts sliding from rest, the total kinetic energy of the sphere at the lowest point $A$ of the bowl will be [given, moment of inertia of sphere $= \frac{2}{5} mr^2$].

$A$ particle is released from height $S$ above the surface of the earth. At a certain height,its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively:

$A$ 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 $.5$)

State the importance of the work-energy theorem. Is the work-energy theorem a scalar or a vector?