$A$ projectile is projected in such a way that it achieves maximum range for a given velocity. Find the velocity of the projectile at its maximum height.

Two particles are simultaneously projected in opposite directions horizontally from a given point in space,where gravity $g$ is uniform. If $u_1$ and $u_2$ are their initial speeds,then the time $t$ after which their velocities are mutually perpendicular is given by

$A$ ball thrown by one player reaches another player in $2 \, s$. The maximum height attained by the ball is ........ $m$. (Take $g = 10 \, m/s^2$)

From the top of a tower $19.6 \,m$ high, a ball is thrown horizontally. If the line joining the point of projection to the point where it hits the ground makes an angle of $45^{\circ}$ with the horizontal, then the initial velocity of the ball is (in $\,m/s$)

In the climax of a movie,the hero jumps from a helicopter and the villain chasing the hero also jumps at the same time from the same level. After some time,when they are at the same horizontal level,the villain fires a bullet horizontally towards the hero. Both the hero and the villain are falling with a constant downward acceleration of $2\ m/s^2$ due to their parachutes. Assuming the hero is within the range of the bullet and air resistance on the bullet is negligible,which of the following is correct?

The maximum height attained by a projectile is increased by $10 \%$ by keeping the angle of projection constant. What is the percentage increase in the time of flight (in $\%$)?