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Chapter 3: Problem 51

As you hurry to catch your flight at the local airport, you encounter a moving walkway that is \(85 \mathrm{m}\) long and has a speed of \(2.2 \mathrm{m} / \mathrm{s}\) relative to the ground. If it takes you \(68 \mathrm{s}\) to cover \(85 \mathrm{m}\) when walking on the ground, how long will it take you to cover the same distance on the walkway? Assume that you walk with the same speed on the walkway as you do on the ground.

Short Answer

Expert verified
It takes approximately 24.64 seconds to cover 85 m on the walkway.

Step by step solution

01

Calculate Your Walking Speed on Ground

First, we need to calculate your walking speed when you are walking on the ground without the assistance of the moving walkway. The speed is given by the formula \( v = \frac{\text{distance}}{\text{time}} \). Here, the distance is 85 m, and the time is 68 s, so your walking speed is \( v = \frac{85}{68} \approx 1.25 \text{ m/s} \).
02

Determine Your Speed on the Walkway

When you walk on the moving walkway, your speed is the sum of your walking speed and the speed of the walkway. Thus, your total speed on the walkway is \( v_{\text{total}} = 1.25 \text{ m/s} + 2.2 \text{ m/s} = 3.45 \text{ m/s} \).
03

Calculate Time to Cover Distance on the Walkway

Now, use the speed on the walkway to find the time it takes to cover the 85 m distance. The formula to calculate time is \( t = \frac{\text{distance}}{\text{speed}} \). Substituting the values, you get \( t = \frac{85}{3.45} \approx 24.64 \text{ s} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relative Speed
Relative speed is a key concept in kinematics that helps understand how objects move in relation to each other. When you walk on a moving walkway, your overall speed is not just the speed at which you walk. You also have to consider the speed of the walkway.
This combined speed is known as the relative speed.
  • Think of the relative speed as your walking speed plus the speed of the walkway.
  • It's important because it represents how fast you are moving in relation to the ground.
  • In the given exercise, the calculation of your relative speed helps determine how much quicker you can travel the 85 m.

By applying the concept of relative speed, we can see that your effective movement is faster, allowing you to travel the same distance more quickly.
Distance-Time Relationship
The distance-time relationship helps us determine how long it takes to cover a distance at a given speed. It's one of the basics of kinematics.

Speed is calculated using the formula:\[ v = \frac{\text{distance}}{\text{time}} \]From this formula, we understand that for a constant speed:
  • The longer the time, the greater the distance you will cover.
  • If you want to reach a specific distance in less time, you need to increase your speed.
The problem illustrates this by comparing how long it takes to travel 85 m on the ground versus on the walkway.
Different times result from different speeds, demonstrating how speed directly changes the time needed to cover distances.
Velocity Addition
Velocity addition is a principle used to calculate the total velocity of an object when there are multiple contributing factors. In this scenario, both your walking speed and the walkway's moving speed combine to give your total velocity.

Here's how to think about velocity addition:
  • When two velocities act in the same direction, you simply add them together.
  • It's like two forces working in unison, making you move faster overall.
Using velocity addition, your walking speed of 1.25 m/s combines with the walkway's 2.2 m/s to become 3.45 m/s.
This insight is pivotal to solving how quickly you cover the 85 m on the walkway, reinforcing the importance of adding velocities to understand total movement.

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Most popular questions from this chapter

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