A ball projected up passes the same height $H$ at 2 s and 10 s . The value of $H$ is (use, $g=9.8 \mathrm{~m} / \mathrm{s}^2$ )

Two towns $X$ and $Y$ are connected by a regular bus service. A bus leaves in either direction at every $t=T$ minutes. A man moving with same speed in the direction $X$ to $Y$ find that a bus goes past him every $t=t_1$ minutes in the direction of his motion, and every $t=t_2$ minutes in the opposite direction. Then, $T$ is given by

A rocket moves straight upward with zero initial velocity and with an acceleration $20 \mathrm{~m} / \mathrm{s}^2$. It runs out of fuel and stops accelerating at the end of 5th second. It reaches a maximum height and falls back to the earth. The speed when it hits the ground is (take $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A particle moves along a straight line, such that its displacement $x$ varies with time $t$ as $x=\alpha t^3+\beta t^2+\gamma$, where $\alpha, \beta$ and $\gamma$ are constants, $v_1$ is the average velocity of the particle during its journey between $t=1 \mathrm{~s}$ and $t=3 \mathrm{~s} . v_2$ is the instantaneous velocity of the particle at $t=3 \mathrm{~s}$. The ratio $\frac{v_1}{v_2}$ is