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?
- A(v₁ + v₂)/2
- B√(v₁v₂)
- C(v₁² + v₂²)/(v₁ + v₂)
- D2v₁v₂/(v₁ + v₂)
Show answer and explanation
Correct answer: (d) 2v₁v₂/(v₁ + v₂)
Explanation
Average velocity = total displacement ÷ total time.
- total displacement = x + x = 2x;
- time for the first part = x/v₁, and for the second = x/v₂;
- total time = x/v₁ + x/v₂ = x(v₁ + v₂)/(v₁v₂).
So average velocity = 2x ÷ [x(v₁ + v₂)/(v₁v₂)] = 2v₁v₂/(v₁ + v₂).
This is the harmonic mean of the two velocities. The simple average (v₁ + v₂)/2 applies only when equal times, not equal distances, are spent at each velocity.