A $100 \mathrm{kg}$ gun fires a ball of 1kg horizontally from a cliff of height $500 \mathrm{m}$. It falls on the ground at a distance of $400 \mathrm{m}$ from the bottom of the cliff. Find the recoil velocity of the gun. (acceleration due to gravity$=10 \mathrm{m} {\mathrm{s}}^{-2}$)

Figure $\left(a\right)$ and $\left(b\right),$ shows $(x, t),(y, t)$diagram of a particle moving in $2$ dimensions If the particle has a mass of $500\mathrm{g}$, find the force (direction and magnitude) acting on the particle.

A person in an elevator accelerating upwards with an acceleration of $2 \mathrm{m} {\mathrm{s}}^{-1}$, tosses a coin vertically upwards with a speed of $20 \mathrm{m} {\mathrm{s}}^{-1}$. After how much time will the coin fall back into his hand? $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

 Figure $\left(a\right)$and $\left(b\right)$ shows $\left(v_x, t\right)$ and $\left(v_y, t\right)$ diagrams for a body of unit mass. Find the force, as a function of time.

 A racing car travels on a track (without banking) $ABCDEFA$ (Figure). $ABC$ is a circular arc of radius $2R$. $CD$ and $FA$ are straight paths of length $R$ and $DEF$ is a circular arc of radius $R=100 m$. The coefficient of friction on the road is $\mu =0.1$. The maximum speed of the car is $50 {\text{ms}}^{-1}$. Find the minimum time for completing one round.

The displacement vector of a particle of mass $m$ is given by $\overrightarrow{r}\left(t\right)=\hat{i}A\cos \omega t+\hat{j}B\sin \omega t$

Show that the trajectory is an ellipse.

The displacement vector of a particle of mass $m$ is given by

$\overrightarrow{r}\left(t\right)=\hat{i}A\cos \omega t+\hat{ȷ}B\sin \omega t$

Show that $\overrightarrow{F}=-m{\omega }^2\overrightarrow{r}$