Ram moves in the east direction at a speed of $6 \, m/s$ and Shyam moves $30^{\circ}$ east of north at a speed of $6 \, m/s$. The magnitude of their relative velocity is ........ $m/s$.

Two cars are moving in the same direction with the same speed $30 \, km/hr$. They are separated by a distance of $5 \, km$. The speed of a car moving in the opposite direction,if it meets these two cars at an interval of $4 \, minutes$,will be.......$km/hr$.

$A$ passenger sitting in a train $A$ moving at $90\,km/h$ observes another train $B$ moving in the opposite direction for $8\,s$. If the velocity of the train $B$ is $54\,km/h$,then the length of train $B$ is $..........\,m$.

Two towns $A$ and $B$ are connected by a regular bus service with a bus leaving in either direction every $T$ min. $A$ man cycling with a speed of $20 \,km/h$ from $A$ to $B$ notices that a bus travelling in the direction of his motion goes past him every $18 \,min$ and every $6 \,min$ he notices a bus travelling in the opposite direction go past him. Assuming that the buses travel with a constant speed,find $T$ and the constant speed of the buses.

$A$ man walking briskly in rain with speed $v$ must slant his umbrella forward making an angle $\theta$ with the vertical. $A$ student derives the following relation between $\theta$ and $v: \tan \theta = v$ and checks that the relation has a correct limit: as $v \rightarrow 0, \theta \rightarrow 0,$ as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not,guess the correct relation.

Two cars, at a certain instant, are $50 \ km$ apart on a line running from south to north. The one farther north is moving west at $25 \ km/hr$. The other is moving towards north at $25 \ km/hr$. How long do they take to reach their distance of closest approach (in $min$)?