Two objects are projected at an angle $\theta^{\circ}$ and $(90-\theta)^{\circ}$ to the horizontal with the same speed. The ratio of their maximum vertical heights is

If two stones are projected at angles $\theta$ and $(90^{\circ}-\theta)$ respectively with the horizontal with a speed of $20 \ ms^{-1}$. If the second stone rises $10 \ m$ higher than the first stone,then the angle of projection $\theta$ is (acceleration due to gravity $g = 10 \ ms^{-2}$) (in $^{\circ}$)

$A$ particle moves in the $X-Y$ plane under the influence of a force such that its linear momentum is $\vec{p}(t)=A[\hat{i} \cos (kt)-\hat{j} \sin (kt)]$,where $A$ and $k$ are constants. The angle between the force and the momentum is (in $^{\circ}$)

Three point particles $P, Q, R$ move in a circle of radius $r$ with different but constant speeds. They start moving at $t = 0$ from their initial positions as shown in the figure. The angular velocities (in $rad/s$) of $P, Q,$ and $R$ are $5\pi, 2\pi,$ and $3\pi$ respectively, in the same sense. Find the time interval after which they are at the same angular position.

Two particles are projected from the same point with the same speed at different angles $\theta _1$ and $\theta _2$ to the horizontal. They have the same range. Their times of flight are $t_1$ and $t_2$ respectively.

$A$ particle moves around a circular path of radius $r$ with uniform speed $V$. After moving half the circle,the average acceleration of the particle is