With what velocity should a ball be projected vertically so that the distance covered by it in the $5^{th}$ second is twice the distance it covers in its $6^{th}$ second? (Take $g = 10 \, m/s^2$)

$A$ tennis ball is released from a height $h$ and after freely falling on a wooden floor,it rebounds and reaches a height $\frac{h}{2}$. The velocity versus height of the ball during its motion may be represented graphically by (graphs are drawn schematically and not to scale):

$A$ ball is thrown up vertically with a certain velocity so that it reaches a maximum height $h$. Find the ratio of the times at which it is at height $\frac{h}{3}$ while going up and coming down respectively.

$A$ body dropped from the top of a tower covers a distance $7x$ in the last second of its journey,where $x$ is the distance covered in the first second. How much time does it take to reach the ground?

$A$ ball is dropped downwards. After $1$ second,another ball is dropped downwards from the same point. What is the distance between them after $3$ seconds?

$A$ body thrown vertically upwards from the ground reaches a maximum height $H$. The ratio of the velocities of the body at heights $\frac{3H}{4}$ and $\frac{8H}{9}$ from the ground is