Two particles projected from the same point with the same speed $u$ at angles of projection $\alpha$ and $\beta$ strike the horizontal ground at the same point. If $h_1$ and $h_2$ are the maximum heights attained by the projectile, $R$ is the range for both and $t_1$ and $t_2$ are their time of flights respectively then,

The velocity of a projectile at the initial point $A$ is $\left(2\hat{\mathrm{i}}-3\hat{\mathrm{j}}\right) \mathrm{m} {\mathrm{s}}^{-1}$. Its velocity (in $\mathrm{m} {\mathrm{s}}^{-1}$) at point $B$ is

In a circus, a performer throws an apple towards a hoop held at $45 \mathrm{m}$ height by another performer standing on a high platform (see figure below). The thrower aims for the hoop and throws the apple with a speed of $24 \mathrm{m} {\mathrm{s}}^{-1}$. At the exact moment that the thrower released the apple, the other performer drops the hoop. The hoop falls straight down. At what height above the ground does the apple go through the hoop?

Two long parallel plates $A$ and $B$ are separated by a distance of $4 \mathrm{cm}$ with an electric field of $45.5 \mathrm{V} {\mathrm{m}}^{-1}$ between the plates directed normally from plate $A$ to plate $B,$ as shown in the figure. An electron is projected from plate $A$ with velocity $v$ at an angle of $30°$ with the surface of plate $A$. The maximum value of $v$ so that the electron does not hit plate $B$ is
$($Assume gravity free space, charge of electron $=1.6\times {10}^{-19}\mathrm{C}$ and mass of electron $\left.=9.1\times {10}^{-31} \mathrm{kg}\right)$

A projectile thrown with an initial velocity of $10 \mathrm{m} {\mathrm{s}}^{-1}$ at an angle $\alpha$ with the horizontal, has a range of $5 \mathrm{m}.$ Taking $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and neglecting air resistance, what will be the estimated value of $\alpha ?$