$A$ man swimming downstream overtakes a float at a point $M$. After travelling a distance $D$,he turns back and passes the float at a distance of $D/2$ from the point $M$. The ratio of the speed of the swimmer with respect to still water $(v_s)$ to the speed of the river $(v_r)$ is:

Two towns $X$ and $Y$ are connected by a regular bus service. $A$ bus leaves in either direction at every $T$ minutes. $A$ man moving with some speed in the direction $X$ to $Y$ finds that a bus goes past him every $t_1$ minutes in the direction of his motion, and every $t_2$ minutes in the opposite direction. Then $T$ is given by

$A$ swimmer wants to cross a $200 \ m$ wide river which is flowing at a speed of $2 \ m/s$. The velocity of the swimmer with respect to the river is $1 \ m/s$. How far from the point directly opposite to the starting point does the swimmer reach the opposite bank (in $m$)?

$A$ 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 $\,m$)?

Two cars $A$ and $B$ are moving with the same speed of $45\,km/h$ along the same direction. If a third car $C$ coming from the opposite direction with a speed of $36\,km/h$ meets the two cars in an interval of $5\,min$,the distance of separation between the two cars $A$ and $B$ should be.......$km$.

$A$ man standing on a road holds his umbrella at $30^\circ$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10 \, km/h$. He finds that raindrops are hitting his head vertically. The speed of raindrops with respect to the moving man will be: