Question

A "moving sidewalk" in an airport terminal moves at 1.0 m/s and is 35.0 m long. If a woman steps on at one end and walks at 1.5 m/s relative to the moving sidewalk, how much time does it take her to reach the opposite end if she walks (a) in the same direction the sidewalk is moving? (b) In the opposite direction?

Short answer

(a) 14.0 s; (b) 70.0 s

Step-by-step solution

Understand the Problem

We are given the speed of a moving sidewalk and the speed of a person walking on it. The sidewalk is 35.0 m long. We need to find the time taken for the woman to reach the opposite end in two scenarios: walking in the same direction and in the opposite direction as the sidewalk.

Determine Effective Speed for (a)

When the woman walks in the same direction as the moving sidewalk, her effective speed is the sum of her walking speed and the speed of the sidewalk. Let: - Speed of woman walking relative to sidewalk = 1.5 m/s - Speed of sidewalk = 1.0 m/s The effective speed, therefore, is:\[ v = 1.5 + 1.0 = 2.5 \, \text{m/s} \]

Calculate Time for (a)

To find the time taken to cross 35.0 meters at the effective speed of 2.5 m/s:Use the formula:\[ \text{time} = \frac{\text{distance}}{\text{speed}} = \frac{35.0}{2.5} = 14.0 \, \text{s} \]

Determine Effective Speed for (b)

For walking in the opposite direction, her effective speed is the difference between her walking speed and the sidewalk speed.Effective speed is:\[ v = 1.5 - 1.0 = 0.5 \, \text{m/s} \]

Calculate Time for (b)

To find the time taken to cross 35.0 meters at the effective speed of 0.5 m/s:Use the formula:\[ \text{time} = \frac{\text{distance}}{\text{speed}} = \frac{35.0}{0.5} = 70.0 \, \text{s} \]

Key concepts

Kinematics

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause this motion. It's all about understanding how an object moves in terms of its speed, velocity, and acceleration. In this exercise, we analyze how a woman's motion is influenced by a moving sidewalk at an airport by using the principles of kinematics.

In our scenario, the key kinematic concept is relative velocity. This describes how different frames of reference can affect perceived speeds. When the woman is on the moving sidewalk, her speed relative to the ground combines with the speed of the sidewalk itself. This is an example of how kinematics helps us predict where and how fast an object will move depending on various relative speeds.

Using kinematic equations, we can determine the effective speed of the woman in two cases: when she walks in the same direction as the moving sidewalk, and when she walks in the opposite direction. This distinction is crucial in solving problems related to kinematic motion. By adding or subtracting her walking speed relative to the sidewalk's speed, we comprehend her movement in a real-world scenario.

Speed Calculation

Speed calculation is essential in determining how long it takes to travel a certain distance. It involves dividing the total distance traveled by the speed to find the time taken. This concept is applied in the given problem by using a straightforward formula:\[\text{time} = \frac{\text{distance}}{\text{speed}}\]When the woman walks in the same direction as the sidewalk, her effective speed becomes the sum of her speed and the sidewalk's speed. This tells us how fast she is moving with respect to a stationary observer, like a person watching her from the terminal:

• Her speed relative to the sidewalk: 1.5 m/s
• Sidewalk speed: 1.0 m/s
• Effective speed: 2.5 m/s
• Time taken to cross: 14.0 s

In contrast, when walking against the sidewalk's motion, her effective speed is her walking speed minus the sidewalk speed, showing the decrease in speed relative to the ground.

• Effective speed: 0.5 m/s
• Time taken: 70.0 s

These calculations illustrate how the same physical concepts can be applied in different situations to yield different outcomes.

Motion in One Dimension

Motion in one dimension refers to movement along a straight line. This type of motion simplifies calculations because you only need to consider a single direction, either forward or backward. In our exercise, the woman is moving along a straight path from one end of the sidewalk to the other, making it a perfect example of one-dimensional motion.

In a one-dimensional scenario, the total displacement is straightforward: it's simply the length of the path, which in this case is 35.0 meters. Factors like acceleration are not a concern here because both the woman and the sidewalk are moving at constant speeds.

This exercise also highlights how the concepts of kinematics and speed calculation come together in one-dimensional motion problems. By dividing the known distance by her effective speed, we were able to find out how quickly she travels that distance depending on her direction relative to the sidewalk's movement. It's an elegant instance of how simple one-dimensional motion concepts can solve seemingly complex scenarios. By applying the basic principles appropriately, you can gain a deeper understanding of motions you encounter in daily life.