$A$ man running at a speed of $5 \text{ km/h}$ finds that the rain falls vertically. When he stops running,he finds that the rain is falling at an angle of $60^{\circ}$ with the horizontal. The velocity of rain with respect to the running man is

$A$ river $200 \,m$ wide is flowing at a rate of $3.0 \,m/s$. $A$ boat is sailing at a velocity of $15 \,m/s$ with respect to the water in a direction perpendicular to the river. How far from the point directly opposite to the starting point does the boat reach on the opposite bank (in $\,m$)?

$A$ river is flowing from west to east at a speed of $8\,m/min$. $A$ man on the south bank of the river,capable of swimming at $20\,m/min$ in still water,wants to swim across the river in the shortest time. He should swim in a direction

Two towns $A$ and $B$ are connected by a regular bus service with a bus leaving in either direction every $T$ minutes. $A$ man cycling with a speed of $20 \; km/h$ in the direction $A$ to $B$ notices that a bus goes past him every $18 \; min$ in the direction of his motion,and every $6 \; min$ in the opposite direction. What is the period $T$ (in $min$) of the bus service and with what speed (assumed constant) do the buses ply on the road?

$A$ particle is moving along the $x$-axis with its coordinate with time '$t$' given by $x(t) = 10 + 8t - 3t^2$. Another particle is moving along the $y$-axis with its coordinate as a function of time given by $y(t) = 5 - 8t^3$. At $t = 1 \; s$,the speed of the second particle as measured in the frame of the first particle is given as $\sqrt{v}$. Then $v$ (in $m^2/s^2$) is:

$A$ person running horizontally observes that rain is falling on his head vertically with a speed of $10\,m/s$. When he stops,he observes that the rain is falling at an angle of $30^o$ with the vertical. Find the speed with which the man was running.