A toy train moves due north at a constant speed 2 m/s along a straight track which is parallel to the wall of a room. The wall is to the east of the track at a distance 4 m. There is a toy dart gun on the train with its barrel fixed in a plane perpendicular to the motion of the train. The gun points at an angle 60^o to the horizontal. There is a vertical line drawn on the wall, stretching from floor to ceiling, and the dart gun is fired at the instant when the line is due east of the gun. If the dart leaves the gun at speed 8 m/s relative to the gun, find the distance by which the dart misses the vertical line. That is, find how far north or south of the vertical line is the point at which the dart hits the wall.

Difficult

Solution

Consider east as x, north as y and vertically upward as z velocity of dart w.r.t. to train at firing
$\begin{array}{l} {\vec{u}}_{dt} = 8 \cos 60^{\circ} \hat{i} + 8 \sin 60^{\circ} \hat{k} \\ = 4 \hat{i} + 4 \sqrt{3} \hat{k} \end{array}$
Velocity of dart w.r.t. ground at firing
${\vec{u}}_{d} = {\vec{u}}_{d t} + {\vec{v}}_{T} = 4 \hat{i} + 4 \sqrt{3} \hat{k} + 2 \hat{j}$
$Time taken to strike wall t = \frac{d}{4} = 1 \sec$
$Displacement along y = 2 \times t = 2 m (North)$

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