A car moves along a straight road, covering a displacement x with a constant velocity v1 and then another equal displacement x with a constant velocity v2. What is the average velocity of the car for the entire journey?

Correct option is D

The correct answer is (d) (2 v1 v2) / (v1 + v2)
Explanation: • Average Velocity is defined as the Total Displacement divided by the Total Time taken for the journey. • The total displacement is 2x (first displacement x + second displacement x). • Time taken for the first part (x) = x / v1 and time taken for the second part (x) = x / v2. • Total Time (T) = x / v1 + x / v2   = x (v1 + v2) / (v1 v2). • Average Velocity (v_avg) = Total Displacement / Total Time   = 2x / [x / v1 + x / v2]   = 2x / {x (v1 + v2) / (v1 v2)}. • Upon simplifying, x cancels out, resulting in the formula:    Average Velocity = (2 v1 v2) / (v1 + v2). This is the harmonic mean of the two velocities.
Information Booster: • This specific formula applies only when the displacements are equal. If the car travels for two equal time intervals with different velocities, the average velocity would be the arithmetic mean:    (v1 + v2) / 2. • According to NCERT Physics (Class 11), average velocity is a vector quantity, but since the car moves on a straight road in one direction, its magnitude is equal to the average speed.
Additional Knowledge:
(a) (v1 + v2) / 2 (Option a) • This is the Arithmetic Mean. It represents the average velocity only when the object moves with different velocities for equal intervals of time.
(b) √(v1 v2) (Option b) • This is the Geometric Mean. It is not typically used to calculate average velocity in standard kinematic problems involving constant velocities over segments.
(c) (v1² + v2²) / (v1 + v2) (Option c) • This is an incorrect mathematical expression for this physical scenario and does not correspond to any standard average velocity derivation.