JEE PYQ: Motion in a Straight Line - Question ID e9871c567d20 (JEE Main 2021)

In problems involving relative motion on an escalator (or any moving platform), the key concept is the addition of velocities when two motions occur simultaneously in the same direction.

• Stationary escalator: When the boy walks on a stationary escalator, his speed relative to the ground is simply his walking speed, $v_b$.
• Moving escalator (boy stationary): When the escalator moves and the boy stands still, the escalator’s speed relative to the ground is $v_e$.
• Combined motion: When the boy walks on the moving escalator, his speed relative to the ground becomes the sum of his walking speed and the escalator’s speed: $v_{\text{total}} = v_b + v_e$.

The fundamental relation connecting speed, distance, and time is: $v = \frac{d}{t}$ where $d$ is the length of the escalator (constant in all scenarios).

Let $d$ be the length of the escalator.

1. Boy walks on stationary escalator: $v_b = \frac{d}{t_1}$
2. Escalator moves, boy stationary: $v_e = \frac{d}{t_2}$
3. Boy walks on moving escalator: The combined speed is: $v_{\text{total}} = v_b + v_e = \frac{d}{t_1} + \frac{d}{t_2}$ The time taken, $t$, is: $t = \frac{d}{v_{\text{total}}} = \frac{d}{\frac{d}{t_1} + \frac{d}{t_2}} = \frac{1}{\frac{1}{t_1} + \frac{1}{t_2}}$ Simplify the denominator: $\frac{1}{t_1} + \frac{1}{t_2} = \frac{t_2 + t_1}{t_1 t_2}$ Thus: $t = \frac{1}{\frac{t_1 + t_2}{t_1 t_2}} = \frac{t_1 t_2}{t_1 + t_2}$

This matches option C.

Students often confuse the addition of times instead of speeds. A frequent mistake is to assume the total time is $t_1 + t_2$ or $\frac{t_1 + t_2}{2}$, which are incorrect because speeds add, not times. Always remember: when two motions occur simultaneously in the same direction, their speeds add, and the time is derived from the combined speed.