Multiple choice

If the body covers one-third distance at speed $v_{1}$, next one third at speed $v_{2}$ and last one third at speed $v_{3}$, then average speed will be

  1. $\dfrac {v_{1}v_{2} + v_{2}v_{3} + v_{3}v_{1}}{v_{1} + v_{2} + v_{3}}$
  2. $\dfrac {v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2}v_{3} + v_{3}v_{1}}$
  3. $\dfrac {v_{1} + v_{2} + v_{3}}{3}$
  4. $\dfrac {3v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2}v_{3} + v_{3}v_{1}}$
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D Correct answer
Explanation

Average speed is total distance divided by total time. Let total distance be 3d. Time taken for each segment is d/v1, d/v2, and d/v3. Total time = d(1/v1 + 1/v2 + 1/v3) = d(v2v3 + v1v3 + v1v2)/(v1v2v3). Average speed = 3d / [d(v1v2 + v2v3 + v3v1)/(v1v2v3)] = 3v1v2v3 / (v1v2 + v2v3 + v3v1).

AI explanation

Average speed is calculated as total distance divided by total time. Assuming a total distance of 3x, each segment is x units long, so the total time is x/v1 + x/v2 + x/v3. The average speed is 3x divided by this sum. Finding a common denominator of v1v2v3, the expression becomes 3xv1v2v3 divided by x(v1v2 + v2v3 + v3v1), which simplifies to 3v1v2v3 divided by (v1v2 + v2v3 + v3v1).

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