A boy reaches the airport and finds that the escalator is not working. He walks up the stationary es

Q: Why do we add the velocities $v_1$ and $v_2$?

Because both the boy and the escalator are moving in the same direction. The boy's movement is relative to the escalator, and the escalator is moving relative to the ground. To find the boy's speed relative to the ground, we add them up.

Q: Does the length of the escalator matter?

No, it doesn't! As you can see in the final calculation, the length $L$ cancels out. The final time only depends on $t_1$ and $t_2$.

Q: What if the boy was walking down the moving escalator?

If he was walking down while the escalator was moving up, his velocity relative to the ground would be $v_2 - v_1$ (assuming the escalator is faster). The time taken would be $t = \frac{t_1 t_2}{t_1 - t_2}$.

Q: Can I just add the times $t_1$ and $t_2$?

No, you cannot add times directly. You must add the rates (velocities) because they are happening simultaneously. Adding times would imply doing one task after the other.

Q: Is this formula similar to anything else in physics?

Yes! It looks exactly like the formula for the equivalent resistance of two resistors in parallel, or the equivalent capacitance of two capacitors in series.

Question

A boy reaches the airport and finds that the escalator is not working. He walks up the stationary escalator in time $t_1$. If he remains stationary on a moving escalator, then the escalator takes him up in time $t_2$. The time taken by him to walk up on the moving escalator will be

Options:

(A) $\frac{t_1 t_2}{t_2 - t_1}$

(B) $\frac{t_1 + t_2}{2}$

(C) $\frac{t_1 t_2}{t_2 + t_1}$

(D) $t_2 - t_1$

Sigma Percentile · Accepted Answer

### The Escalator Dilemma: A Lesson in Relative Motion

Imagine you are rushing through an airport, luggage in hand, only to find that the escalator is broken. You have no choice but to walk up the stationary steps. Let's say this takes you a time $t_1$.

Now, imagine a different scenario. You are exhausted, but thankfully, the escalator is working. You simply stand on it, and it carries you to the top in time $t_2$.

But what if you are in a real hurry? What if you walk up the escalator *while* it is moving? How long will that take? This classic physics problem is a beautiful illustration of relative velocity.

### Analyzing the Setup

To solve this, we need to define a few variables. Let the total length of the escalator be $L$.

In the first scenario, you are walking up a stationary escalator. Your walking speed, which we will call $v_1$, is simply the total distance divided by the time it takes. So, $v_1 = \frac{L}{t_1}$.

In the second scenario, you are standing still on a moving escalator. The speed of the escalator itself, let's call it $v_2$, is the total distance divided by the time it takes. So, $v_2 = \frac{L}{t_2}$.

### The Master Equation

Now comes the crucial part. When you walk on a moving escalator, your total velocity relative to the ground is the sum of your walking speed and the escalator's speed.

This is the core principle of relative motion in one dimension. Your resultant velocity $v$ is given by:

$$v = v_1 + v_2$$

### Final Calculation

Let's substitute the expressions we found for $v_1$ and $v_2$ into our master equation:

$$v = \frac{L}{t_1} + \frac{L}{t_2}$$

We can factor out the length $L$ and find a common denominator to simplify this expression:

$$v = L \left( \frac{t_1 + t_2}{t_1 t_2} \right)$$

Finally, we want to find the total time $t$ it takes to reach the top. We know that time is distance divided by velocity. So, $t = \frac{L}{v}$.

Substituting our expression for $v$ into this equation, we get:

$$t = \frac{L}{L \left( \frac{t_1 + t_2}{t_1 t_2} \right)}$$

The length $L$ beautifully cancels out, leaving us with our final answer:

**$$t = \frac{t_1 t_2}{t_1 + t_2}$$**

This elegant result shows that the time taken is independent of the length of the escalator. It only depends on the individual times $t_1$ and $t_2$.

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Visualized Solution

$Visualizing the Escalator$

$Case 1: Stationary Escalator$

$Case 2: Moving Escalator$

$Case 3: Walking on Moving Escalator$

$Substituting Velocities$

$Simplifying the Expression$

$Calculating Total Time$

$The Way Forward$

The Sigma Insight: Relative Velocity

The Escalator Dilemma

Analyzing the Setup

The Master Equation

Final Calculation

Similar Questions

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Visualizing the Escalator

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Case 1: Stationary Escalator

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Case 2: Moving Escalator

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Case 3: Walking on Moving Escalator

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Substituting Velocities

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Simplifying the Expression

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating Total Time

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Way Forward