Question

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 4m. 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 ${60}^{\circ }$ 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.

• A. 3 m north
• B. 2 m south
• C. 2 m north
• D. 3 m south

Answer 1

The correct option is C

2 m north

In this point of view, the train will be moving in the direction of the y-axis which will be perpendicular to the plane of your screen and going through it.

$\therefore$ Initial velocity of the dart in the x direction is $ucos\theta$

$\therefore$ $u_x$ = $ucos\theta$, $u_2$ = $usin\theta$, $u_y$ = $2\frac{m}{s}$

We know horizontal displacement is $4m$.

$\therefore$ $x=\left(ucos\theta \right)t$ (as there is no acceleration along x-direction)

$\Rightarrow$$4$ = $8\left(cos60\right)t$

$\Rightarrow$$4=8\times \frac{1}{2}t$

$\Rightarrow$$t=1s$

$\therefore$It reaches the wall in $1s$. We need to find its displacement along the y - direction. Again no acceleration along y-direction

$\therefore$$y=u_y\times t$

= $2\times 1$

= $2m$

$\therefore$The dart will hit $2m$ north of the line.