Two Questions on Relative Velocity

The relative velocity of a body A with respect to a body B is given by

v = v_A – v_B

To make the symbol of the relative velocity ‘v’ more informative, it is usually written as v_AB so that the above equation becomes

v_AB = v_A – v_B

Since the velocity is a vector, you have to find the vector difference v_A – v_B to get the relative velocity.

Now, consider the following MCQ:

A river is flowing with a velocity 1.5 î – ĵ with respect to the ground. A boat is moving with a velocity 2.5 î + 2 ĵ with respect to the ground where î and ĵ are unit vectors in the X and Y directions respectively. The relative velocity of the boat with respect to water in the river is

(a) 4 î + 3 ĵ (b) 4 î + ĵ (c) – î + 3 ĵ (d) î – 3 ĵ (e) î + 3 ĵ

To obtain the relative velocity of the boat with respect to water, you have to subtract (vectorially) the velocity of flow of the river from the velocity of the boat. Therefore, the relative velocity is (2.5 î + 2 ĵ) – (1.5 î – ĵ) = î + 3 ĵ

Here is a simple question which appeared in the Kerala Engineering entrance 2007 question paper:

Two trains are moving with equal speed in opposite directions along two parallel 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 1:2, then the speed of each train must be

(a) 3 u (b) 2 u (c) 5 u (d) 4 u (e) u

If ‘v’ is the speed of each train, the relative velocities of the trains with respect to the wind are v – u and v + u. (One train is moving along the direction of the wind and the other is moving opposite to the wind).

Therefore we have (v – u)/( v + u) = ½, so that v = 3u.