On a long horizontal belt,a child runs to and fro with a speed of $9\,km/h$ (with respect to the belt) between his father and mother located $50\,m$ apart on the moving belt. The belt moves with a speed of $4\,km/h.$ For an observer on the platform,the speed of the child in the direction of motion of the belt is.........$km/h$.

$A$ boy runs on a horizontal road with a speed of $4 \ m \ s^{-1}$ while it is raining. He sees that the rain is making an angle $\theta$ with the vertical while running from west to east. However, when he runs from east to west, the angle is $\alpha$. The rain is pouring down at an angle $45^{\circ}$ with the vertical and at a speed of $8 \ m \ s^{-1}$ as shown in the figure. The ratio $\frac{\tan \theta}{\tan \alpha}$ is,

$A$ shell is fired at a speed of $200\ m/s$ at an angle of $37^o$ above the horizontal from the top of a tower $80\ m$ high. At the same instant,another shell is fired from a jeep travelling away from the tower at a speed of $10\ m/s$ as shown. The velocity of this shell relative to the jeep is $250\ m/s$ at an angle of $53^o$ with the horizontal. Find the time (in $sec$) taken by the two shells to come closest.

Wind is blowing in the north direction at a speed of $2 \, m/s$,which causes the rain to fall at some angle with the vertical. With what velocity should a cyclist drive so that the rain appears vertical to him?

$A$ man can swim with a speed of $4.0 \; km/h$ in still water. How long does he take to cross a river $1.0 \; km$ wide if the river flows steadily at $3.0 \; km/h$ and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank?

On a long horizontally moving belt,a child runs to and fro with a speed $9 \; km \; h^{-1}$ (with respect to the belt) between his father and mother located $50 \; m$ apart on the moving belt. The belt moves with a speed of $4 \; km \; h^{-1}$. For an observer on a stationary platform outside,what is the:
$(a)$ speed of the child running in the direction of motion of the belt?
$(b)$ speed of the child running opposite to the direction of motion of the belt?
$(c)$ time taken by the child in $(a)$ and $(b)$?
Which of the answers alter if motion is viewed by one of the parents?