Exercises

A ball is projected horizontally from an incline to strike a cart sliding on the incline. Neglect height of cart and point of projection of ball above incline. At the instance the ball is thrown, the speed of the cart is $v$ (in $\mathrm{m} {\mathrm{s}}^{-1}$). Find $v$ so that the ball strikes the cart.

A glass windscreen whose inclination with the vertical can be changed is mounted on a car. The car moves horizontally with a speed of $2 \mathrm{m} {\mathrm{s}}^{-1}.$ Now the windscreen is placed at the angle $\alpha$ with the vertical such that the raindrops falling vertically downwards with velocity $6 \mathrm{m} {\mathrm{s}}^{-1}$ strike the windscreen perpendicularly. What is the value of $\tan \alpha$?

Two particles are projected simultaneously from two points $O$ and $O\text{'}$ such that $10 \mathrm{m}$ is horizontal and $5 \mathrm{m}$ is the vertical distance between them as shown in the figure. They are projected at the same inclination $60°$ to the horizontal with the same velocity $10 \mathrm{m} {\mathrm{s}}^{-1}.$ Find the time after which their separation becomes minimum.

A particle is moving in $x-y$ plane. At a certain instant of time, the components of its velocity and acceleration are as follows: $v_x=3 \mathrm{m} {\mathrm{s}}^{-1}, v_y=4 \mathrm{m} {\mathrm{s}}^{-1}, a_x=2 \mathrm{m} {\mathrm{s}}^{-2}$ and $a_y=1 \mathrm{m} {\mathrm{s}}^{-2}.$ Find the rate of change of speed (in $\mathrm{m} {\mathrm{s}}^{-2}$) at this moment.

A ball is projected at an angle $\mathrm{\theta }.$ As the ball flies through the air, the observer at the point of projection, follows it with his eyes. When it reaches maximum height $Y,$ his eyes are directed at an angle $\mathrm{\varphi }$ with respect to the horizontal. What is the value of $\frac{\tan \left(\mathrm{\theta }\right)}{\tan \left(\mathrm{\varphi }\right)}?$

A ball leaves a horizontal table with velocity $v_0=5 \mathrm{m} {\mathrm{s}}^{-1}.$ The ball bounces elastically from a vertical wall at a horizontal distance $D\left(=8 \mathrm{m}\right)$ from the table, as shown in figure. The ball then strikes the floor a distance $x_0$ from the table $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right).$ Find the value of $x_0$ (in $\mathrm{m}$).

Two particles are projected simultaneously from two points $O$ and $O\text{'}$ such that $10 \mathrm{m}$ is the horizontal and $5 \mathrm{m}$ is the vertical distance between them as shown in the figure. They are projected at the same inclination $60°$ to the horizontal with the same velocity $10 \mathrm{m} {\mathrm{s}}^{-1}$. Find the time (in sec) after which their separation becomes minimum.

The windscreens of two motorcars are having slopes $30°$ and $15°$ respectively. At what ratio $\frac{v_1}{v_2}$ of the velocities of cars will their drivers see the hailstones bounced by windscreen of their cars in the vertical direction? Assume hailstones are falling vertically.