$A$ car moving at a speed $v$ is stopped by a retarding force $F$ in a distance $s$. If the retarding force were $3 F$,then the car will be stopped in a distance

Derive the work-energy theorem for a variable force exerted on a body in one dimension.

$A$ ball is moving to and fro about the lowest point $A$ of a smooth hemispherical bowl. If it is able to rise up to a height of $20 \,cm$ on either side of $A$,its speed at $A$ must be .......... $m/s$ (Take $g = 10 \,m/s^2$,mass of the body = $5 \,g$).

$A$ block of mass $\sqrt{2} \, kg$ is released from the top of an inclined smooth surface as shown in the figure. If the spring constant of the spring is $100 \, N/m$ and the block comes to rest after compressing the spring by $1 \, m$,then the total distance travelled by the block along the incline before it comes to rest is ......... $m$.

In the given figure, the block of mass $m$ is dropped from the point $A$. The expression for the kinetic energy of the block when it reaches point $B$ is ................

The rain drop of mass $1 \text{ g}$, starts with zero velocity from a height of $1 \text{ km}$. It hits the ground with a speed of $5 \text{ m/s}$. The work done by the unknown resistive force is . . . . . . $J$. (take $g = 10 \text{ m/s}^2$)