Two stones are projected with the same speed but making different angles with the horizontal. Their ranges are equal. If the angle of projection of one is $\pi / 3$ and its maximum height is $h_1$,then the maximum height of the other will be:

The position vector of a particle is given as $\vec{r} = (t^2 - 8t + 12)\hat{i} + t^2\hat{j}$. The time after which the velocity vector and acceleration vector become perpendicular to each other is equal to ........ $sec$.

An object is moving in a circle of radius $100 \, m$ with a constant speed of $31.4 \, m/s$. What is its average speed for one complete revolution?

There are two points $P$ and $Q$ on a projectile path with velocities $v_P$ and $v_Q$ respectively,such that $v_P$ is perpendicular to $v_Q$. If $\alpha$ is the angle that $v_P$ makes with the horizontal at point $P$,find the correct option.

Twelve persons are initially at $12$ corners of a regular polygon of $12$ sides of side length $a$. Each person moves with a uniform speed $v$ in such a manner that person $1$ is always directed towards person $2$,person $2$ towards person $3$,person $3$ towards person $4$,and so on. The time after which they meet is

$A$ cyclist starts from the centre $O$ of a circular park of radius $1 \text{ km}$,reaches the edge $P$ of the park. Then,the cyclist cycles along the circumference to point $Q$ and returns to the centre along $QO$ as shown in the figure. If the round trip takes $10 \text{ minutes}$,the net displacement and average speed of the cyclist (in metre and kilometre per hour respectively) are: