Preeti reached the metro station and found that the escalator was not working. She walked up the stationary es

Question

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time t₁. On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time t₂. The time taken by her to walk upon the moving escalator will be:

Accepted Answer

Let the length of the escalator be $L$.
1.  **Velocity of Preeti ($v_p$):** When walking on the stationary escalator, $v_p = \frac{L}{t_1}$.
2.  **Velocity of the Escalator ($v_e$):** When standing on the moving escalator, $v_e = \frac{L}{t_2}$.
3.  **Relative Velocity:** When she walks on the moving escalator, her net velocity ($v$) with respect to the ground is the sum of her walking speed and the escalator's speed (since both are in the same direction) .
    $$ v = v_p + v_e = \frac{L}{t_1} + \frac{L}{t_2} $$
    $$ v = L \left( \frac{1}{t_1} + \frac{1}{t_2} \right) = L \left( \frac{t_1 + t_2}{t_1 t_2} \right) $$
4.  **Calculate Time ($t$):** The time taken to cover distance $L$ with velocity $v$ is:
    $$ t = \frac{L}{v} = \frac{L}{L \left( \frac{t_1 + t_2}{t_1 t_2} \right)} $$
    $$ t = \frac{t_1 t_2}{t_1 + t_2} $$

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