$A$ body is projected with a velocity $(\hat{i} + 2\hat{j}) \text{ ms}^{-1}$,where $\hat{i}$ is along the horizontal and $\hat{j}$ is vertically upward. Then the equation of its trajectory is $(g = 10 \text{ ms}^{-2})$.

$A$ projectile can have the same range $R$ for two angles of projection. If $t_1$ and $t_2$ are the times of flight in the two cases,then their product is:

$A$ projectile can have the same range for two angles of projection. If $h_1$ and $h_2$ are the maximum heights when the range in the two cases is $R$,then the relation between $R$,$h_1$,and $h_2$ is:

$A$ boy can throw a stone up to a maximum height of $10 \ m$. The maximum horizontal distance that the boy can throw the same stone up to will be .......... $m$.

In a spring gun having a spring constant $k = 100\, \text{N/m}$,a small ball $B$ of mass $m = 100\, \text{g}$ is placed in its barrel (as shown in the figure) by compressing the spring by $x = 0.05\, \text{m}$. $A$ box is placed at a distance $d$ on the ground so that the ball falls into it. If the ball leaves the gun horizontally at a height of $h = 2\, \text{m}$ above the ground,find the value of $d$ in meters. (Take $g = 10\, \text{m/s}^2$)

Two balls with same mass and initial velocity are projected at different angles in such a way that the maximum height reached by the first ball is $8$ times higher than that of the second ball. If $T_1$ and $T_2$ are the total flight times of the first and second ball,respectively,then the ratio of $T_1$ to $T_2$ is: