Question

A bird flies to and fro between two cars which move with velocities ${v_1}$ and ${v_2}$ . If the speed of the bird is ${v_3}$ and the initial distance of separation between them is d, find the total distance covered by the bird till the cars meet.

Answer 1

Hint: The time taken by the bird is obtained on dividing distance by speed (relative speed) and the required distance travelled by the bird is calculated by the product of speed of the bird and time taken by it.

Complete step by step answer:
 A bird flies to and fro between two cars i.e., one car is moving from the left side (A) and another car is moving from the right side (B) and the bird flies from car A to car B and again from car B to car A. The velocity at which car A is moving is ${v_1}$ and velocity at which car B is moving is ${v_2}$ . The speed at which bird is flying is ${v_3}$ . The initial distance between the two cars is d.
The relative velocity is defined as the velocity at which car A is moving with respect to car B and also the velocity at which car B is moving with respect to car A. When the two objects are moving in the same direction, relative velocity is the difference of the individual velocities and when they are moving in an opposite direction, the relative velocity is the sum of the velocity of first and second object.
As we know, both the cars are moving in an opposite direction, the relative velocity $\left( {{V_R}} \right) = {v_1} + {v_2}$ and the time taken by the bird to cover a distance of d be t.
$t = \dfrac{d}{{{v_1} + {v_2}}}\left[ {time = \dfrac{{displacement}}{{velocity}}} \right]$
The total distance covered by the bird till the cars meet,
$D = {v_3} \times t\left[ {displacement = velocity \times time} \right]$
$\implies D = {v_3} \times \left( {\dfrac{d}{{{v_1} + {v_2}}}} \right)$
$\therefore D = \dfrac{{{v_3}d}}{{{v_1} + {v_2}}}$ .

Note:
The velocity at which the two cars are moving is calculated from relative velocity. In this case, both the cars are moving in an opposite direction i.e., car A is moving from left side and car B is moving from right side and the relative velocity is the sum of their individual velocities.