Problem 83 - College Physics

Question

A shopper in a mall is on an escalator that is moving downward at an angle of \(41.8^{\circ}\) below the horizontal at a constant speed of \(0.75 \mathrm{~m} / \mathrm{s}\). At the same time a little boy drops a toy parachute from a floor above the escalator and it descends at a steady vertical speed of \(0.50 \mathrm{~m} / \mathrm{s}\). Determine the speed of the parachute toy as observed from the moving escalator.

Step-by-Step Solution

Answer

The parachute appears stationary relative to the shopper on the escalator.

Step 1: Understanding the problem

We need to find the speed of the toy parachute as observed by the shopper on the escalator. The key here is to find the relative speed of the parachute with respect to the moving escalator.

Step 2: Identify given parameters

The shopper and the escalator move at a constant speed of 0.75 m/s downward at an angle of 41.8° below the horizontal. The parachute descends with a steady vertical speed of 0.50 m/s.

Step 3: Calculate escalator's vertical speed component

The vertical speed component of the escalator's motion is found using the formula: \[v_{e,y} = v_e \sin(\theta)\]where \(v_e = 0.75 \text{ m/s}\) and \(\theta = 41.8^{\circ}\) so \[v_{e,y} = 0.75 \sin(41.8^{\circ})\approx 0.50 \text{ m/s}\]

Step 4: Determine relative speed of parachute as seen by shopper

Since both the parachute and the vertical component of the escalator's speed are the same (0.50 m/s), the parachute's speed relative to the escalator is essentially zero in the vertical direction.

Key Concepts

Escalator Physics

When considering escalators in physics, it's important to understand how they move. An escalator is essentially a moving staircase, and it typically moves at a constant speed at a specific angle relative to the horizontal axis. This means that the escalator has both vertical and horizontal components of motion.

• The angle of movement gives the escalator a distinct path, often measured in degrees from the horizontal.
• The constant speed of the escalator is a crucial factor since it determines how swiftly a person or object traveling with it moves both vertically and horizontally.

When analyzing objects or people on an escalator, these angles and speeds must be known to predict motion accurately. In the case of our shopper, we know the escalator is moving downwards at an angle of 41.8° with a speed of 0.75 m/s. Understanding this helps in determining how other objects, like the parachute, will appear to move relative to the shopper.

Velocity Components

To comprehend motion on an escalator, we break down velocities into their components. Any motion at an angle can be decomposed into horizontal and vertical components using trigonometry. Here’s how this works:

• The horizontal component: It describes how fast something moves parallel to the ground. It's determined using the cosine of the angle.
• The vertical component: It represents how fast the object moves up or down, calculated using the sine of the angle.

For the escalator:

• The vertical component of its velocity is calculated as: \(v_{e,y} = v_e \sin(41.8^\circ)\).
• Given that \(v_e = 0.75 \text{ m/s}\), substituting the values gives a vertical component of approximately 0.50 m/s.

This component signifies how fast the escalator carries something downwards. In parallel, the parachute descends vertically at a speed of 0.50 m/s, matching the vertical speed of the escalator's motion.

Relative Speed Calculation

Relative motion is observing how fast an object moves compared to another moving object. In this scenario, we find the relative speed of the parachute as seen by the shopper.

• Since the escalator's vertical speed component and the parachute's descending speed are both 0.50 m/s, they effectively cancel each other out in the vertical direction.
• This means that, to the shopper, the parachute would appear to have no speed vertically, as if it's stationary in regards to the vertical motion.

By considering such components, it becomes clear that relative speed calculations rely on comparing motion parameters from different perspectives. Here, the shopper's reference frame makes it seem as if the parachute is not moving vertically at all, due to the matching speeds. Understanding this can simplify complex movement problems significantly, allowing for intuitive assessment of how objects interact in motion.