A person travelling in a straight line moves with a constant velocity v₁ for a certain distance x and with a c

Question

A person travelling in a straight line moves with a constant velocity v₁ for a certain distance x and with a constant velocity v₂ for the next equal distance. The average velocity v is given by the relation:

Accepted Answer

1.  **Definition of Average Velocity:** Average velocity is defined as the total displacement divided by the total time interval .
    $$ v_{avg} = \frac{	ext{Total Displacement}}{	ext{Total Time}} $$
2.  **Calculate Time Intervals:**
    *   For the first half distance $x$, velocity is $v_1$. Time taken $t_1 = \frac{x}{v_1}$.
    *   For the second half distance $x$, velocity is $v_2$. Time taken $t_2 = \frac{x}{v_2}$.
3.  **Calculate Total Values:**
    *   Total Displacement $= x + x = 2x$.
    *   Total Time $= t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2} = x \left( \frac{1}{v_1} + \frac{1}{v_2} ight)$.
4.  **Derive Relation:**
    $$ v = \frac{2x}{x(\frac{1}{v_1} + \frac{1}{v_2})} $$
    $$ v = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} $$
    Rearranging the terms:
    $$ \frac{1}{v} = \frac{\frac{1}{v_1} + \frac{1}{v_2}}{2} $$
    $$ \frac{2}{v} = \frac{1}{v_1} + \frac{1}{v_2} $$

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