Q.At a metro station, a girl walks up a stationary escalator in time $t_1$. If she remains stationary on the escalator, then the escalator take her up in time $t_2$. The time taken by her to walk up on the moving escalator will be

Concept: Relative velocity – when the girl walks on the moving escalator, her effective speed is the sum of her walking speed and the escalator's speed.

Let the length of the escalator be $L$.

Step 1: The girl's walking speed (escalator stationary) is $v_g = \frac{L}{t_1}$.

Step 2: The escalator's speed (girl stationary) is $v_e = \frac{L}{t_2}$.

Step 3: When she walks up the moving escalator, both speeds add:

$v_{\text{total}} = v_g + v_e = \frac{L}{t_1} + \frac{L}{t_2} = L\left(\frac{t_1 + t_2}{t_1 t_2}\right)$ …

The key insight is that speeds add when motions are in the same direction. When you walk on a moving walkway, you cover ground faster than either you or the walkway alone would manage. The natural language here is rates: if the girl climbs at a certain rate (escalator-lengths per unit time) and the escalator moves at its own rate, the combined rate is simply the sum.

Let the length of the escalator be $L$.

When the girl walks up the stationary escalator in time $t_1$, her walking speed is

When she stands still and the escalator carries her up in time $t_2$, the escalator's speed is

Now consider what happens when both move together.

1. The effective speed is the sum of the two speeds. The girl walks at $v_{\text{girl}}$ relative to the escalator, and the escalator itself moves at $v_{\text{esc}}$ relative to the ground. Relative to the ground, her speed is

2. Factor out the common length. …

Concept: Combined-Rate Reasoning, Verified by Limiting Cases

Method: Rate-Addition by Analogy, Checked at the Extremes (a "two workers" style argument, sanity-checked rather than re-derived from scratch)

This treats the problem as a "combined rate" problem — the same structure as two people working together, or two pipes filling a tank — and, crucially, verifies the resulting formula by checking it against physically obvious extreme cases, rather than only deriving it once through algebra.

Steps

1. Express each rate as "escalator-lengths per unit time." If the escalator has length $L$: the girl's own walking rate (escalator off) is $\dfrac{1}{t_1}$ (in units of $L$ per unit time), and the escalator's own rate (girl standing still) is $\dfrac{1}{t_2}$.

2. Combined rates add, exactly as with two workers or two taps filling the same tank together:

3. Time is the reciprocal of the combined rate (to cover one full escalator length, $L=1$ in these units):

4. Sanity-check by taking $t_1\to\infty$ (the girl walks so slowly it's as if she isn't walking at all): the formula should reduce to "just ride the escalator," i.e. $t\to t_2$.

5. Sanity-check by taking $t_2\to\infty$ (the escalator is effectively stationary): the formula should reduce to "just walk," i.e. $t\to t_1$. $\lim_{t_2\to\infty}\frac{t_1t_2}{t_1+t_2} = \lim_{t_2\to\infty}\frac{t_1}{1+t_1/t_2} = t_1 \quad\checkmark$ …