A ball is thrown onto a rough floor with speed $u$ at angle $\theta =45°$. If it rebounds with speed $v$ at the same angle $\varphi =45°$. Find the coefficient of kinetic friction $\left(\mu \right)$ between the floor and the ball. The coefficient of restitution, $e=0.6$

Sand is falling on a flat car being pulled with constant speed. The rate of mass falling on the cart is constant. Then the horizontal component of force exerted by the falling sand on the cart,

A ball is thrown horizontally from a height with a certain initial velocity at time $t=0$. The ball bounces repeatedly from the ground with the coefficient of restitution less than $1$ as shown.

Neglect air resistance and taking the upward direction as positive, which figure qualitatively depicts the vertical component of the ball's velocity $\left(v_{\mathrm{y}}\right)$ as a function of time $\left(t\right)?$ 

A uniform square wooden sheet of side a has its centre of mass located at point $O$ as shown in the figure on the left. A square portion of side $b$ of this sheet is cut out to produce and $\mathrm{L}$-shaped sheet as shown in the figure on the right.

The centre of mass of the L-shaped sheet lies at the point $P$ (in the diagram) when,

Two uniform plates of the same thickness and area but of different materials, one shaped like an isosceles triangle and the other shaped like a rectangle are joined together to form a composite body as shown in the figure. If the centre of mass of the composite body is located at the midpoint of their common side, the ratio between masses of the triangle to that of the rectangle is,