$A$ particle is moving in a circle. Which of the following statements is correct?

$A$ ball of mass $m$ is thrown vertically upward. Another ball of mass $2m$ is thrown at an angle $\theta$ with the vertical. Both the balls stay in the air for the same period of time. The ratio of the maximum heights attained by the two balls respectively is $\frac{1}{x}$. The value of $x$ is $.....$

$A$ player kicks a football at an angle of $30^{\circ}$ with the horizontal with an initial speed of $30 \,ms^{-1}$. $A$ second player, standing at a distance of $21 \sqrt{3} \,m$ from the first player in the direction of the kick, starts running to catch the ball at the same instant it is kicked. What is the minimum speed of the second player to catch the ball before it hits the ground? (Take acceleration due to gravity $g = 10 \,ms^{-2}$)

Two objects of equal mass are projected with the same velocity $v$ at the same angle $\theta$ towards each other from a distance $R$. What is the momentum of the system at the moment of collision?

The ranges and heights for two projectiles projected with the same initial velocity at angles $42^{\circ}$ and $48^{\circ}$ with the horizontal are $R_{1}, R_{2}$ and $H_{1}, H_{2}$ respectively. Choose the correct option:

$A$ particle moving in a circle of radius $R$ with a uniform speed takes a time $T$ to complete one revolution. If this particle were projected with the same speed at an angle $\theta$ to the horizontal,the maximum height attained by it equals $4R$. The angle of projection,$\theta$,is then given by