Problem 74 - College Physics

Question

A moving walkway in an airport is \(75 \mathrm{~m}\) long and moves at a speed of \(0.30 \mathrm{~m} / \mathrm{s}\). A passenger, after traveling \(25 \mathrm{~m}\) while standing on the walkway, starts to walk at a speed of \(0.50 \mathrm{~m} / \mathrm{s}\) relative to the surface of the walkway. How long does she take to travel the total distance of the walkway?

Step-by-Step Solution

Answer

145.83 seconds

Step 1: Calculate time standing still

Calculate the time the passenger spends standing on the walkway while covering the initial 25 meters. The walkway moves at 0.30 m/s, so use the formula for time, \(t = \frac{d}{v}\), where \(d\) is distance and \(v\) is speed: \( t_1 = \frac{25}{0.30} \approx 83.33 \text{ seconds}.\)

Step 2: Determine remaining distance and speed walking

Once the passenger starts walking, she walks the remaining distance on the walkway. The total length of the walkway is 75 meters, so the remaining distance is \(75 - 25 = 50\) meters. Her walking speed relative to the walkway is 0.50 m/s, and the walkway itself moves at 0.30 m/s, giving her a combined speed of \(0.50 + 0.30 = 0.80\) m/s relative to the ground.

Step 3: Calculate time walking

Now, calculate the time to cover the remaining 50 meters while walking. The formula for time is again used: \( t_2 = \frac{50}{0.80} = 62.5 \text{ seconds}.\)

Step 4: Total time to travel the walkway

Add the time spent standing and the time spent walking to find the total time taken to travel the entire length of the walkway: \( t_{total} = t_1 + t_2 = 83.33 + 62.5 = 145.83 \text{ seconds}.\)

Key Concepts

Kinematics

Understanding the concept of kinematics can help greatly when solving problems involving motion. Kinematics focuses on the geometry of motion, in terms of position, velocity, and acceleration, without considering the forces that cause the motion. In this problem, we need to determine how long the passenger travels along a moving walkway. We need to understand two kinds of positions or locations involved: when the passenger is standing still and when she is moving relative to the walkway.

When calculating time, we rely on the fundamental kinematic equation relating time (\(t\)), distance (\(d\)), and speed (\(v\)):

• \(t = \frac{d}{v}\)

Initially, as the passenger stands still on the walkway, only the speed of the walkway affects her motion. The key is to separate each segment of motion: first when she stands and second when she walks.

Kinematics allows us to decipher motion clearly by breaking it into manageable parts and using simple mathematics to identify each duration.

Relative Velocity

Relative velocity is an important concept when analyzing scenarios where two speeds must be considered simultaneously. In the exercise, the passenger walks on a moving walkway, so her speed relative to the fixed ground is not the same as her speed relative to the walkway.

To find her speed from the perspective of an observer standing on the ground, we must add her walking speed to the speed of the walkway:

• \(\text{Relative velocity} = \text{Passenger's speed relative to walkway} + \text{Walkway's speed}\)
• \(0.50 \text{ m/s} + 0.30 \text{ m/s} = 0.80 \text{ m/s}\)

Knowing how to calculate relative velocity helps to ensure we are considering all parts of motion in the correct context.

This concept is not only useful in physics problems involving motion on walkways but also in many real-world scenarios, such as when determining the impact of current on a boat's speed or the wind on an airplane.

Motion on a Walkway

Motion on a moving walkway requires understanding both individual movement and interaction with the walkway's motion. The exercise shows how, when standing still, the passenger still moves due to the walkway's speed.

At first, the passenger stands on the walkway, moving at the walkway's speed. It's essential to calculate the time spent in this phase by dividing the distance (25 m) by the walkway's velocity. Once she begins walking, both her velocity and the walkway's velocity combine to provide a new total speed, resulting in a quicker traversal of the remaining distance (50 m).

This motion is typical of scenarios where two speeds combine. Here’s how it is practically applied:

• The moving walkway helps passengers travel faster and more comfortably by adding its speed to the passenger's.
• The walkway effectively decreases the time needed to travel the entire length, illustrating an efficient mode of transportation.

Understanding motion on a walkway can also aid in planning experiences, such as minimizing travel time or understanding how quickly someone can move from one airport terminal to another.