Question

You are walking on a moving walkway in an airport. The length of the walkway is \(59.1 \mathrm{~m}\). If your velocity relative to the walkway is \(2.35 \mathrm{~m} / \mathrm{s}\) and the walkway moves with a velocity of \(1.77 \mathrm{~m} / \mathrm{s}\), how long will it take you to reach the other end of the walkway?

Short answer

Answer: The person takes approximately 14.34 seconds to reach the other end of the walkway.

Step-by-step solution

Find the overall velocity of the person relative to the ground

To get the overall velocity of the person relative to the ground (v), we add their velocity relative to the walkway (v_w) to the walkway's velocity (v_m). v = v_w + v_m Given: v_w = 2.35 m/s v_m = 1.77 m/s Now, we can calculate the overall velocity, v: v = 2.35 + 1.77 = 4.12 m/s

Calculate the time it takes for the person to reach the other end of the walkway

To find the time (t) it takes for the person to reach the other end of the walkway, we can use the formula: t = distance / velocity Given: distance = 59.1 m velocity = 4.12 m/s Now, we can calculate the time: t = 59.1 / 4.12 ≈ 14.34 s So, it would take approximately 14.34 seconds for the person to reach the other end of the walkway.

Key concepts

Velocity

Velocity is a fundamental concept in physics that represents the speed of an object in a given direction. It is a vector quantity, which means it includes both a magnitude (how fast the object is moving) and a direction (where the object is heading).

In our exercise, you encounter two velocities: the velocity of the person relative to the walkway (\(v_w\text{ = 2.35 m/s}\)), and the velocity of the moving walkway itself (\(v_m\text{ = 1.77 m/s}\)). When the velocities of the person and the walkway are in the same direction, they add up to give the person's overall velocity relative to the ground. This is crucial for determining how quickly you can cover a certain distance, in this case, the length of the walkway.

The formula for computing velocity is given as \(v = \frac{\text{distance}}{\text{time}}\), which we can rearrange to find time when distance and velocity are known: \(t = \frac{\text{distance}}{v}\). Therefore, having a clear understanding of velocity allows you to solve problems involving motion, such as calculating the time needed to traverse a particular distance.

Frame of Reference

The frame of reference is an abstract concept used to describe the position and movement of objects. It is essentially a set of coordinates or a point of view from which the motion is observed and measured. In physics problems, specifying the frame of reference is critical because the description of motion can be quite different depending on the observer's viewpoint.

In the context of our exercise, we can consider multiple frames of reference. One is the frame of reference of the walkway, and the other is the frame of reference of the ground or the airport itself. The velocity of the person walking (\(v_w\text{ = 2.35 m/s}\)) is measured relative to the walkway. However, to find the overall velocity of the person which is used to calculate the travel duration, we must shift to the ground's frame of reference and then combine the person's velocity with the walkway's velocity (\(v_m\text{ = 1.77 m/s}\)).

Understanding frames of reference can dramatically improve your comprehension of relative motion, as it explains why different observers may report different velocities for the same moving object.

Motion Kinematics

Motion kinematics is the branch of mechanics that deals with the description of motion without considering the forces that cause it. It includes concepts like displacement, velocity, acceleration, and time. Kinematic equations enable us to predict the future position of objects or to postulate their previous positions given certain information.

Kinematics is at the core of our exercise, where we analyze the movement of an individual on a walkway. Using the kinematic equation \(t = \frac{\text{distance}}{\text{velocity}}\), we assess how long it takes for the person to travel the entire length of the walkway (\(59.1\text{m}\)). The equation challenges us to apply our understanding of velocity and motion within a particular frame of reference, solidifying the importance of kinematics in solving real-world problems related to motion.

By dissecting and solving exercises like the one provided, students improve their kinematic problem-solving abilities, which are essential for understanding more complex concepts in physics.