Question

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

It takes about 24.64 seconds to cover 85 meters on the walkway.

Step-by-step solution

Determine Walking Speed on Ground

To find out how fast you walk on the ground, divide the distance traveled by the time taken. The formula to use is: \[v = \frac{d}{t} \]where \( d = 85 \ \text{m} \) and \( t = 68 \, \text{s} \). Substitute the given values:\[v = \frac{85}{68} \approx 1.25 \, \text{m/s}\]So, your walking speed on the ground is approximately \(1.25 \, \text{m/s}\).

Calculate Combined Speed on Walkway

On the walkway, your speed combines the speed of the walkway and your walking speed. Using the formula:\[v_{\text{combined}} = v_{\text{walkway}} + v_{\text{self}}\]Substitute \(v_{\text{walkway}} = 2.2 \, \text{m/s} \) and \(v_{\text{self}} = 1.25 \, \text{m/s}\):\[v_{\text{combined}} = 2.2 + 1.25 = 3.45 \, \text{m/s}\]Your speed on the walkway is \(3.45 \, \text{m/s}\).

Calculate Time to Cover Distance on Walkway

Calculate the time it takes to cover the \(85 \, \text{m}\) on the walkway using the combined speed. Use the formula:\[t = \frac{d}{v}\]where \(d = 85 \, \text{m}\) and \(v = 3.45 \, \text{m/s}\):\[t = \frac{85}{3.45} \approx 24.64 \, \text{s}\]It takes about \(24.64 \, \text{s}\) to cover \(85 \, \text{m}\) on the walkway.

Key concepts

Relative Velocity

Relative velocity is a core concept in kinematics. It helps us understand how the speed of an object appears to change when observed from different frames of reference.
In our airport exercise, the relative velocity concept allows us to understand how your walking speed is altered by the addition of the walkway's speed. Think of it like this: when you're walking alone, your speed is relative to the ground, but when you step onto the moving walkway, the velocity of the walkway gets added to your personal walking speed.
In mathematical terms, this is represented by the formula: \( v_{\text{combined}} = v_{\text{walkway}} + v_{\text{self}} \).

• \( v_{\text{walkway}} \) is the speed of the walkway relative to the ground.
• \( v_{\text{self}} \) is your speed on the ground.
• \( v_{\text{combined}} \) is the total speed you'll experience on the walkway.

This combined speed is crucial for calculating how quickly you cover the walkway's length.

Speed Calculation

Calculating speed is an essential part of solving motion-related problems. Speed is defined as the distance traveled divided by the time taken. For our scenario, you first calculate your walking speed on the ground.
By using the formula \( v = \frac{d}{t} \), we found your walking speed on the ground to be approximately \(1.25\,\text{m/s}\).
To determine how fast you'll move on the moving walkway, you have to compute the combined speed. The combined speed is the sum of the walkway's speed and your walking speed.

• The formula \( v_{\text{combined}} = v_{\text{walkway}} + v_{\text{self}} \) lets you calculate this.
• Replacing the values, you get \( 2.2\,\text{m/s} + 1.25\,\text{m/s} = 3.45\,\text{m/s} \).

This speed tells us how quickly you move relative to the ground when on the walkway.

Motion on a Moving Walkway

When you're on a moving walkway, your motion is affected by two components: your walking speed and the walkway's speed. This combination allows you to cover distances more swiftly than walking on flat ground alone.
This scenario reflects real-life situations like moving sidewalks in airports. To find out how long it takes to cover a set distance on the walkway, use the combined speed to calculate time.
The formula \( t = \frac{d}{v} \) helps you compute this time.

• Inserting \( d = 85\,\text{m} \) and \( v = 3.45\,\text{m/s} \) gives \( t = \frac{85}{3.45} \approx 24.64\,\text{s} \).
• Here, the result shows that the time required to traverse the 85 meters on the moving walkway is around 24.64 seconds.

Understanding this combination of speeds explains why moving walkways make travel faster and can be advantageously used in large areas like airports.