$A$ particle is projected at an angle $\theta$ with the horizontal from the ground. The slope $(m)$ of the trajectory of the particle varies with time $(t)$ as:

Galileo,in his book $Two \text{ } new \text{ } sciences$,stated that "for elevations which exceed or fall short of $45^{\circ}$ by equal amounts,the ranges are equal". Prove this statement.

$A$ particle covers $50\, m$ distance when projected with an initial speed. On the same surface,it will cover a distance of ......... $m$ when projected with double the initial speed.

$A$ heavy particle is projected from a point on the horizontal at an angle $60^\circ$ with the horizontal with a speed of $10 \ m/s$. Then the radius of curvature of its path at the instant of crossing the same horizontal will be ......... $m$.

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:

The ceiling of a long hall is $25\; m$ high. What is the maximum horizontal distance that a ball thrown with a speed of $40\; m/s$ can go without hitting the ceiling of the hall (in $; m$)?