A ball of mass $1 \mathrm{kg}$ moving with a velocity of $5 \mathrm{m}/\mathrm{s}$ collides elastically with rough ground at an angle $\theta$ with the vertical as shown in figure. What can be the minimum coefficient of friction if ball rebounds vertically after collision? (given $\tan \theta =2$)

Three balls $A, B$ and $C\left(m_A=m_C=4m_B\right)$ are placed on a smooth horizontal surface. Ball $B$ collides with ball $C$ with an initial velocity $v$ as shown in figure. Find the total number of collisions between the balls (all collisions are elastic).

A particle with a mass of $1 \mathrm{kg}$ is having a velocity of $10 \mathrm{m}/\mathrm{s}$ in +ve x -direction at $t=0.$ Forces ${\overrightarrow{F}}_1$ and ${\overrightarrow{F}}_2$ act on the particle whose magnitudes are changing with time according to the variation shown in figure. The magnitude of the velocity of the particle at $t=3\mathrm{s}$ (neglect gravity effect) is found to be $n\sqrt{5} \mathrm{m}/\mathrm{s}.$. Find the value of $n.$

Two identical smooth balls are projected from points O and A on the horizontal ground with same speed of projection. The angle of projection in each case is $30°$ (see figure). The distance between O and A is $100 \mathrm{m}.$ The balls collide in mid-air and return to their respective points of projection. If the coefficient of restitution is $0.7,$ find the speed of projection of either ball (in $\mathrm{m}/\mathrm{s}$) correct to nearest integer. (Take $g=10 \mathrm{m}{ \mathrm{s}}^{-2}$ and $\sqrt{3}=1.7$)

A smooth ball of mass $1 \mathrm{kg}$ is projected with velocity $7 \mathrm{m} {\mathrm{s}}^{-1}$ horizontal from a tower of height $3.5 \mathrm{m}.$ It collides elastically with a wedge of mass $3 \mathrm{kg}$ and inclination of $45°$ kept on ground (figure). The ball collides with the wedge at a height of $1 \mathrm{m}$ above the ground. Find the velocity of the wedge (in $\mathrm{m} {\mathrm{s}}^{-1}$) after collision. (Neglect friction at any contact.)

A small ball is projected from point P towards a vertical wall as shown in figure. It hits the wall when its velocity is horizontal. Ball reaches point P after one bounce on the floor. The coefficient of restitution assuming it to be same for two collisions is $n/2.$ All surfaces are smooth. Find the value of $n.$

A small steel ball A is suspended by an inextensible thread of length $l=1.5 \mathrm{m}$ from O. Another identical ball is thrown vertically downwards such that its surface remains just in contact with thread during downward motion and collides elastically with the suspended ball. If the suspended ball just completes vertical circle after collision, calculate the velocity (in $\mathrm{cm}/\mathrm{s}$) of the falling ball just before collision $\left(g=10 \mathrm{m}/{\mathrm{s}}^2\right)$