Multiple choice

A particle travels half of total distance with speed $v_1$ and next half with speed $v_2$ along a straight line. Find out the average speed of the particle?

  1. $\displaystyle \frac{2v_1v_2}{v_1\, +\, v_2}$
  2. $\displaystyle \frac{2v_1v_2}{v_1\, -\, v_2}$
  3. $\displaystyle \frac{v_1\, +\, v_2}{2v_1v_2}$
  4. $\displaystyle \frac{v_1\, -\, v_2}{2v_1v_2}$
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A Correct answer
Explanation

Average speed is total distance divided by total time. Let total distance be 2d. Time taken for first half = d/v1. Time taken for second half = d/v2. Total time = d/v1 + d/v2 = d(v1+v2)/(v1*v2). Average speed = 2d / [d(v1+v2)/(v1*v2)] = 2*v1*v2 / (v1+v2).

AI explanation

For a journey where equal distances are covered at two different speeds, apply the harmonic mean formula: 2xy divided by (x + y). Replacing x with v1 and y with v2 gives the exact average speed for the entire journey. The resulting expression is (2 * v1 * v2) divided by (v1 + v2).

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