Question

Suppose you are riding a bike with a speed of $ 10\,m/s $ due east relative to a person A who is walking on the ground towards east. If your friend B walking on the ground due west measures your speed as $ 15\,m/s $ , find the relative velocity between two reference frames A and B.

Answer 1

Hint : In this solution, we will use the concepts of relative velocity to determine the relative velocity between frames A and B. Since person A and person B are moving in opposite directions, we will select one direction as positive and then find the relative velocity.

Complete step by step answer
We’ve been told that we are riding a bike with a speed of $ 10\,m/s $ due east relative to a person A, who is walking on the ground towards the east and the speed of $ 15\,m/s $ for friend B, who is moving westwards.
Let our velocity on the bike be $ v $ , the velocity of person A be $ {v_A} $ and the velocity of friend B as $ {v_B} $
Since our velocity relative to person A is $ 10\,m/s $ , we can say
 $ \Rightarrow v - {v_A} = 10 $
And for friend B who is moving West in a direction opposite to us,
 $ \Rightarrow v + {v_B} = 15 $
Now we want to find the relative velocity between A and B. Since A and B are moving in opposite directions, their relative velocity will be $ {v_{AB}} = {v_A} + {v_B} $ . To find this value, we will subtract equation (1) from (2) and write
 $ \Rightarrow v + {v_B} - v + {v_A} = 15 - 10 $
 $ \Rightarrow {v_B} + {v_A} = 5\,m/s $
Hence the relative velocity of person A and friend B relative to each other is equal to $ 5\,m/s $ .

Note
While calculating relative velocity, we must be careful in remembering that for objects travelling in the same direction will have their absolute velocities subtracted to find their relative velocity while for objects moving in opposite directions, their absolute velocities will add up to find their relative velocity since they are moving in the opposite direction.