Source: Extra Practice

Rain is falling vertically downwards with a speed of 4 km/h4\text{ km/h}. A girl walks on a straight road with a velocity of 3 km/h3\text{ km/h}. The magnitude of the velocity of the rain relative to the girl is:

Options

Option A is correct

5 km/h5\text{ km/h}

Option B

7 km/h7\text{ km/h}

Option C

1 km/h1\text{ km/h}

Option D

3.5 km/h3.5\text{ km/h}

Explanation

Let the vertical downward direction be along j^-\hat{j} and the horizontal direction of the girl's movement be along i^\hat{i}. Thus, the velocity of the rain is vr=4j^\vec{v}_r = -4\hat{j} and the velocity of the girl is vg=3i^\vec{v}_g = 3\hat{i}. The velocity of the rain relative to the girl is vrg=vrvg=3i^4j^\vec{v}_{rg} = \vec{v}_r - \vec{v}_g = -3\hat{i} - 4\hat{j}. The magnitude of this relative velocity is vrg=(3)2+(4)2=9+16=5 km/h|\vec{v}_{rg}| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = 5\text{ km/h}. Option 2 (7 km/h7\text{ km/h}) and option 3 (1 km/h1\text{ km/h}) are algebraic additions/subtractions which do not apply to perpendicular vector quantities.