Two projectiles,one fired from the surface of the Earth with velocity $10 \, m/s$ and the other fired from the surface of another planet with initial speed $5 \, m/s$,trace identical trajectories. The value of the acceleration due to gravity on the planet is ......... $m/s^2$.

It is possible to project a particle with a given velocity in two possible ways so as to make them pass through a point $P$ at a horizontal distance $r$ from the point of projection. If $t_1$ and $t_2$ are times taken to reach this point in two possible ways,then the product $t_1 t_2$ is proportional to

The maximum height attained by a projectile is increased by $10\,\%$ by increasing its speed of projection,without changing the angle of projection. The percentage increase in the horizontal range will be $...........\,\%$

An object is projected from the ground with speed $u$ at an angle $\theta$ with the horizontal. The radius of curvature of its trajectory at the maximum height from the ground is ..........

$A$ body is projected into a vertical $X-Y$ plane with the $X$-axis along the horizontal and the $Y$-axis along the vertical with an initial velocity $(10 \hat{i} + p \hat{j}) \ m/s$. If the maximum height reached by the body is $50 \%$ of its range,then the value of $p$ is:

The trajectory of a projectile,projected from the ground,is given by $y = x - \frac{x^2}{20}$,where $x$ and $y$ are measured in meters. The maximum height attained by the projectile will be $...........\,m$.