Question

A person walks up a stalled 15 -m-long escalator in \(90 \mathrm{~s}\). When standing on the same escalator, now moving, the person is carried up in \(60 \mathrm{~s}\). How much time would it take that person to walk up the moving escalator? Does the answer depend on the length of the escalator?

Short answer

It will take the person 36 seconds to walk up the moving escalator. The length of the escalator does not affect this time, as the speeds were calculated independently of it.

Step-by-step solution

Determine the person's walking speed

To find the speed of the person on a stalled escalator, divide the length of the escalator (15 m) by the time taken to walk up it (90 s). This gives a speed of \( 15 \, \mathrm{m} / 90 \, \mathrm{s} = 0.167 \, \mathrm{m/s}\).

Determine the escalator's speed

To find the speed of the moving escalator, divide the length of the escalator (15 m) by the time taken to be carried up by it (60 s). This gives a speed of \( 15 \, \mathrm{m} / 60 \, \mathrm{s} = 0.25 \, \mathrm{m/s}\).

Determine the combined speed

By adding the person’s speed when walking (0.167 m/s) to the speed of the escalator (0.25 m/s), the combined speed can be found. So, the combined speed is \( 0.167 \, \mathrm{m/s} + 0.25 \, \mathrm{m/s} = 0.417 \, \mathrm{m/s}\).

Calculate the time taken to walk up the moving escalator

To find the time taken to walk up the moving escalator, divide the length of the escalator (15 m) by the combined speed (0.417 m/s). So, the time taken is \( 15 \, \mathrm{m} / 0.417 \, \mathrm{m/s} = 36 \, \mathrm{s}\).

Key concepts

Relative Speed

When analyzing motion involving more than one moving object, understanding relative speed is crucial. Relative speed refers to how fast one object is moving in relation to another. Consider the case of a person walking on an escalator: both the person and the moving escalator have their own speeds. To solve problems like these, it’s essential to understand how these speeds interact.

The relative speed between two objects moving towards the same point can be found by simply adding their individual speeds. In our example, the speed at which the person moves while walking on the escalator is found by adding their walking speed to the escalator's speed. This combined speed is what we use to determine how quickly the person reaches the top.

By comprehending relative speed, we can effectively analyze how two speeds interact, leading to a better understanding of the resultant speed when two motions are combined.

Walking Speed

Walking speed is the rate at which a person covers a certain distance while walking. It is an essential aspect of understanding kinematics, especially when dealing with mixed motion scenarios such as walking on a moving escalator. In our example, the individual walking speed was calculated as 0.167 m/s.

To calculate walking speed, you divide the distance by the time it takes to walk that distance. This calculation helps us quantify how fast a person is by themselves, without any aid from other moving systems.

• Formula for walking speed: walking speed = distance / time

When combining this speed with other factors like escalator movement, it’s useful to understand the base speed upon which other speeds are added or subtracted.

Motion Analysis

Motion analysis involves studying and breaking down the various components that influence movement. In the context of the problem provided, we looked at both static and dynamic components. First, we analyze the person walking on a stalled escalator to determine their walking speed. Then we analyze the motion of the moving escalator to find its speed.

By dissecting each part of the movement separately, and then combining them for overall speed analysis, we gain a concrete understanding of total motion. This enables us to predict outcomes, such as determining how long it will take someone to walk up a moving escalator.

• Step 1: Determine individual speeds (walking and escalator).
• Step 2: Add these speeds to find the combined speed.
• Step 3: Use the combined speed to calculate time needed for traversing the entire path.

This structured motion analysis illuminates how multiple forces combine to affect an object’s motion, allowing us to solve complex motion problems with confidence.