Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $\overrightarrow{\mathrm{u}}$ and the other from rest with uniform acceleration $\overrightarrow{\mathrm{f}}$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time.
(1) $\frac{u\cos\alpha}{f}$
(2) $\frac{u\sin\alpha}{f}$
(3) $\frac{f\cos\alpha}{u}$
(4) $u\sin\alpha$

Two fixed frictionless inclined plane making an angle $30 ^ { \circ }$ and $60 ^ { \circ }$ with the vertical are shown in the figure. Two block $A$ and $B$ are placed on the two planes. What is the relative vertical acceleration of $A$ with respect to $B$?
(1) $4.9 \mathrm {~ms} ^ { - 2 }$ in horizontal direction
(2) $9.8 \mathrm {~ms} ^ { - 2 }$ in vertical direction
(3) zero
(4) $4.9 \mathrm {~ms} ^ { - 2 }$ in vertical direction

A passenger train of length $60 m$ travels at a speed of $80 \mathrm {~km} / \mathrm { hr }$. Another freight train of length $120 m$ travels at a speed of $30 \mathrm {~km} / \mathrm { hr }$. 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:
(1) $\frac { 5 } { 2 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 11 } { 5 }$
(4) $\frac { 25 } { 11 }$

A passenger sitting in a train $A$ moving at $90 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ observes another train $B$ moving in the opposite direction for 8 s . If the velocity of the train $B$ is $54 \mathrm {~km} \mathrm {~h} ^ { - 1 }$, then length of train $B$ is:
(1) 120 m
(2) 320 m
(3) 80 m
(4) 200 m

Two trains $A$ and $B$ of length $l$ and $4l$ are travelling into a tunnel of length $L$ in parallel tracks from opposite directions with velocities $108 \mathrm{~km~h}^{-1}$ and $72 \mathrm{~km~h}^{-1}$, respectively. If train $A$ takes 35 s less time than train $B$ to cross the tunnel then, length $L$ of tunnel is: (Given $L = 60l$)
(1) 1200 m
(2) 900 m
(3) 1800 m
(4) 2700 m