$A$ passenger train of length $60\, m$ travels at a speed of $80\, km/hr$. Another freight train of length $120\, m$ travels at a speed of $30\, km/hr$. The ratio of times taken by the passenger train to completely cross the freight train when:
$(i)$ they are moving in the same direction,and
$(ii)$ in the opposite directions is

$A$ 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,

$A$ ship is travelling due east at $10 \, km/hr$. $A$ second ship,heading $30^o$ east of north,is always due north of the first ship. The speed of the second ship in $km/hr$ is:

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 $min$)?

Two trains are moving with equal speed in opposite directions along two parallel railway tracks. If the wind is blowing with speed $u$ along the track so that the relative velocities of the trains with respect to the wind are in the ratio of $1:2$,then the speed of each train must be

$A$ ship $X$ moving due north with speed $v$ observes that another ship $Y$ is moving due west with the same speed $v$. The actual velocity of $Y$ is $........$.