For a given initial velocity,a projectile is fired at angles of $30^\circ$ and $60^\circ$. Find the ratio of their maximum heights.

$A$ ball is thrown from the ground at an angle $\theta$ with the horizontal and with an initial speed $u_0$. For the resulting projectile motion,the magnitude of the average velocity of the ball up to the point when it hits the ground for the first time is $V_1$. After hitting the ground,the ball rebounds at the same angle $\theta$ but with a reduced speed of $u_0 / \alpha$. Its motion continues for a long time as shown in the figure. If the magnitude of the average velocity of the ball for the entire duration of motion is $0.8 V_1$,the value of $\alpha$ is:

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 ........ $sec$.

$A$ particle of mass $m$ is projected with velocity $v$ making an angle of $45^\circ$ with the horizontal. When the particle lands on the level ground,the magnitude of the change in its momentum will be

$A$ body of mass $2 \, kg$ has an initial velocity of $3 \, m/s$ along $OE$ and it is subjected to a force of $4 \, N$ in a direction perpendicular to $OE$. The distance of the body from $O$ after $4 \, s$ will be ........... $m$.

Two projectiles are thrown simultaneously in the same plane from the same point. If their velocities are $v_1$ and $v_2$ at angles $\theta_1$ and $\theta_2$ respectively from the horizontal,and $v_1 \sin \theta_1 = v_2 \sin \theta_2$,then choose the incorrect statement.