A body is projected with velocity $v_1$ from the point $A$ as shown in the figure. At the same time, another body is projected vertically upwards from $B$ with velocity $v_2$. The point $B$ lies vertically below the highest point of the first particle. For both the bodies to collide, $\frac{v_2}{v_1}$ should be

A ball is projected from a point $A$ with some velocity at an angle $30°$ with the horizontal as shown in the figure. Consider a target at point $B$. The ball will hit the target if it is thrown with a velocity $v_0$ equal to

Three boys are running on an equitriangular track with the same speed $5 \mathrm{m} {\mathrm{s}}^{-1}$. At the start, they were at the three corners with velocity along indicated directions. The velocity of approach of any one of them towards another at $t=10 \mathrm{s}$ equals

A man holds an umbrella at $30°$ with the vertical to keep himself dry. Then, he runs at a speed of $10 \mathrm{m} {\mathrm{s}}^{-1}$, and find the rain drops to be hitting vertically. Study the following statements and find the correct options.

$\left(\mathrm{i}\right)$ Velocity of rain with respect to Earth is $20 \mathrm{m} {\mathrm{s}}^{-1}$.

$\left(\mathrm{ii}\right)$ Velocity of rain with respect to man is $10\sqrt{3} \mathrm{m} {\mathrm{s}}^{-1}$.

$\left(\mathrm{iii}\right)$ Velocity of rain with respect to Earth is $30 \mathrm{m} {\mathrm{s}}^{-1}$.

$\left(\mathrm{iv}\right)$ Velocity of rain with respect to man is $10\sqrt{2} \mathrm{m} {\mathrm{s}}^{-1}$.

Rain appears to fall vertically to a man walking at $3 \mathrm{km} {\mathrm{h}}^{-1}$, but when he changes his speed to double, the rain appears to fall at $45°$ with vertical. Study the following statements and find which of them are correct.

$\mathrm{i}$. Velocity of rain is $2\sqrt{3} \mathrm{km} {\mathrm{h}}^{-1}$

$\mathrm{ii}$. The angle of fall of rain (with vertical) is $\mathrm{\theta }={\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)$.
$\mathrm{iii}$. The angle of fall of rain (with vertical) is $\text{θ}={\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$.
$\mathrm{iv}$. Velocity of rain is $3\sqrt{2} \mathrm{km} {\mathrm{h}}^{-1}$.

A motorboat is to reach at a point $30°$ upstream on the other side of a river flowing with velocity $5 \mathrm{m} {\mathrm{s}}^{-1}$. The velocity of the motorboat with respect to water is $5\sqrt{3} \mathrm{m} {\mathrm{s}}^{-1}$. The driver should steer the boat at an angle