An airplane airspeed indicator reads $100 \, m/s$ and its compass shows that it is heading $37^{\circ}$ east of north. The meteorological information provided to the navigator is that the wind velocity is $20 \, m/s$ towards east. The speed of the airplane relative to the ground is closest to ............ $\, m/s$.

$A$ person walks up a stalled escalator in $90 \ s$. When standing on the same moving escalator,he reaches the top in $60 \ s$. The time it would take him to walk up the moving escalator will be: (in $s$)

$A$ 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 $m$)?

There are two particles $P$ and $Q$ separated by a distance of $10 \ km$ and moving with velocities of $10 \ kmph$ as shown in the figure. The time elapsed before they reach their minimum separation is.........$hr$.

$A$ stationary man observes that the rain is falling vertically downward. When he starts running with a velocity of $12 \ km/h$,he observes that the rain is falling at an angle of $60^{\circ}$ with the vertical. The actual velocity of the rain is:

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.