Problem 118 - Fundamentals of Physics

Question

An airport terminal has a moving sidewalk to speed passengers through a long corridor. Larry does not use the moving sidewalk; he takes \(150 \mathrm{~s}\) to walk through the corridor. Curly, who simply stands on the moving sidewalk, covers the same distance in \(70 \mathrm{~s}\). Moe boards the sidewalk and walks along it. How long does Moe take to move through the corridor? Assume that Larry and Moe walk at the same speed.

Step-by-Step Solution

Answer

Moe takes approximately 47.73 seconds to move through the corridor.

Step 1: Determine Larry's Speed

We are given that Larry takes 150 seconds to walk through the corridor without the moving sidewalk. Let \( d \) be the distance of the corridor. Larry's speed \( v_L \) is the distance divided by the time he takes, so we have:\[ v_L = \frac{d}{150} \]

Step 2: Determine the Speed of the Moving Sidewalk

Curly takes 70 seconds to travel the same distance by just standing on the moving sidewalk. Therefore, the speed of the moving sidewalk \( v_S \) is:\[ v_S = \frac{d}{70} \]

Step 3: Find Moe's Combined Speed

Moe walks on the moving sidewalk, so his speed will be the sum of Larry's walking speed and the speed of the sidewalk:\[ v_M = v_L + v_S = \frac{d}{150} + \frac{d}{70} \]

Step 4: Calculate Moe's Time to Cover the Distance

Since Moe's speed \( v_M \) is the combined speed of Larry and the sidewalk, we need Moe's time (\( t_M \)) to cover the distance \( d \). Using the formula \( v = \frac{d}{t} \), we find:\[ t_M = \frac{d}{v_M} = \frac{d}{\left(\frac{d}{150} + \frac{d}{70}\right)} \]

Step 5: Simplify Moe's Time

Simplifying Moe's time:First, find a common denominator for the sum of speeds:\[ v_M = \frac{d}{150} + \frac{d}{70} = \frac{7d}{1050} + \frac{15d}{1050} = \frac{22d}{1050} \]\[ t_M = \frac{d}{\left(\frac{22d}{1050}\right)} = \frac{1050}{22} \approx 47.727 \text{ seconds} \]

Step 6: Conclusion

Moe takes approximately 47.73 seconds to move through the corridor.

Key Concepts

Speed Calculation

Speed is one of the foundational concepts you will deal with in relative motion exercises like this one. It can be simply described as how fast something moves. The formula to calculate speed is straightforward:

• Speed = Distance ÷ Time

The idea is to determine how long it takes to cover a specific distance. In the exercise, Larry's speed is calculated using this fundamental formula. He takes 150 seconds to walk through a corridor of distance \(d\). Thus, Larry's speed \(v_L\) is \( \frac{d}{150} \). This concept helps us understand the role speed plays in comparing different modes of travel across the same distance.

Time and Distance

Time and distance are intricately linked in motion problems. Understanding their relationship helps you compare how long it takes different people or objects to cover the same distance. In our example, the distance \(d\) remains constant, whether someone is walking or standing on the moving sidewalk. The time taken varies based on the speed. Larry takes 150 seconds by walking, whereas Curly takes 70 seconds simply standing, using the moving walkway's speed.
By keeping distance constant, it's easier to compare their respective speeds and time taken. This helps in finding out how the addition of two speeds (as seen with Moe) affects the time it takes to traverse the corridor. Moe benefits from both his walking speed and the speed of the moving sidewalk, leading us to the last important concept: velocity addition.

Velocity Addition

Velocity addition is essential in problems involving multiple speeds or motions. When you combine two velocities, like a person walking on a moving sidewalk, you add the velocities together because they are in the same direction. This total speed then gives you a new perspective on the time taken to travel a distance.In our exercise, Moe's combined speed \(v_M\) is the sum of Larry's walking speed \(v_L\) and the speed of the sidewalk \(v_S\):

• \( v_M = v_L + v_S \)

This forms the basis for calculating Moe's actual travel time in the corridor. Using velocity addition simplifies the process of determining the new speed with which he moves, demonstrating how these principles help solve real-world relative motion problems.