A particle moving in a straight line covers half the distance with speed 6 m / s. The other half is covered in two equal time intervals with speeds 9 m / s and 15 m / s respectively. The average speed of the particle during the motion is :

Solution

Physics sir · 4 steps

Equal times means a plain average; equal distances means a harmonic mean. Use each in its own half.

1. Second half first. The second half is covered in two equal time intervals, so its average speed is the plain mean: (9 + 15)/2 = 12 m s⁻¹.
2. Equal distances. Now the whole trip is two equal distances, one at 6 m s⁻¹ and one at 12 m s⁻¹. For equal distances the average speed is the harmonic mean of the two speeds.

   v_(avg) = (2 v₁ v₂)/(v₁ + v₂) = (2 × 6 × 12)/(6 + 12) = 144/18 = 8 m s⁻¹

3. Check with time. Take each half as d. The first half takes d/6 and the second takes d/12, so the total time is d/4. Total distance 2d divided by d/4 gives 8 m s⁻¹.
4. So. The average speed of the particle is 8 m/s, option (d).

The idea to keep. Average speed is always total distance over total time. Equal time intervals give the plain mean of the speeds; equal distances give the harmonic mean 2v₁v₂/(v₁+v₂).

Answer from the NTA final answer key. Solution by PrepWiser.