Problem 82 - University Physics with Modern Physics

Question

You are hungry and decide to go to your favorite neighbor- hood fast-food restaurant. You leave your apartment and take the elevator 10 flights down (each flight is 3.0 \(\mathrm{m}\) ) and then go 15 \(\mathrm{m}\) south to the apartment exit. You then proceed 0.2 \(\mathrm{km}\) east, turn north, and go 0.1 \(\mathrm{km}\) to the entrance of the restaurant. (a) Determine the displacement from your apartment to the restaurant. Use unit vector notation for your answer, being sure to make clear your choice of coordinates. (b) How far did you travel along the path you took from your apartment to the restaurant, and what is the magnitude of the displacement you calculated in part (a)?

Step-by-Step Solution

Answer

Displacement: \(200 \hat{i} + 85 \hat{j} - 30 \hat{k}\) m. Path length: 345 m. Displacement magnitude: ~219.3 m.

Step 1: Understand the Problem

The problem requires us to find two things: 1) the displacement from your starting point to the end location given their paths in unit vector notation, and 2) the total distance traveled and the magnitude of the displacement. Understanding these terms is crucial: displacement is the straight-line distance between two points, while distance is the actual path traveled.

Step 2: Choose and Define the Coordinate System

We will set a coordinate system where the starting point (apartment) is at the origin. Let the positive x-axis go east and the positive y-axis go north, while the positive z-axis goes up. This defines our unit vectors as \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) for east, north, and up, respectively.

Step 3: Calculate Vertical Displacement

From the starting point, you take the elevator down 10 flights. Each flight is 3 meters, so you go down \(10 \times 3 = 30\) meters. Vertical displacement is: \(-30 \hat{k}\) meters.

Step 4: Calculate Horizontal Displacement

First move is 15 meters south, which translates to \(-15 \hat{j}\) meters since going south is opposite the y-axis. Next, you travel 0.2 km east, which is \(0.2 \times 1000 = 200\) meters \(\hat{i}\). Finally, you travel 0.1 km north, which is \(0.1 \times 1000 = 100\) meters \(\hat{j}\). Combine these to get horizontal displacement: \(200 \hat{i} + (100 - 15) \hat{j} = 200 \hat{i} + 85 \hat{j}\) meters.

Step 5: Combine Displacements to Find Total Displacement

Add the vertical and horizontal displacements for the final displacement vector: \(\vec{d} = 200 \hat{i} + 85 \hat{j} - 30 \hat{k}\). This is our displacement in unit vector notation.

Step 6: Compute the Total Distance Traveled

We find the total path length by summing the individual path segments: down the elevator (30 meters), south (15 meters), east (200 meters), and north (100 meters). Total distance = \(30 + 15 + 200 + 100 = 345\) meters.

Step 7: Calculate the Magnitude of the Displacement

The magnitude of a vector \( \vec{d} = a \hat{i} + b \hat{j} + c \hat{k} \) is \( \sqrt{a^2 + b^2 + c^2} \). Thus, the magnitude of displacement is \(\sqrt{200^2 + 85^2 + (-30)^2} = \sqrt{40000 + 7225 + 900} = \sqrt{48125}\). Calculating, we get \(\approx 219.3\) meters.

Key Concepts

Coordinate System

When solving problems involving displacement, it's crucial to define a coordinate system. This helps in understanding and calculating movements in different directions efficiently. The problem states that you start at your apartment, which we set as the origin of our coordinate system. The coordinate system is typically composed of three axes:

• X-axis: Runs east-west. In this problem, east is positive, which aligns with the direction of your travel after leaving the apartment.
• Y-axis: Runs north-south. The north direction is considered positive, opposite to when you first travel 15 meters south.
• Z-axis: Up-down direction. For this problem, moving down the elevator is negative on the z-axis.

By setting these directions, you clearly frame the problem in a three-dimensional space, making it easier to describe your movements with vectors. This clear setup is key to calculating the needed displacements accurately.

Vector Notation

Vectors provide a powerful way to describe movement and direction within a defined coordinate system. Unlike simple numbers, vectors have both magnitude and direction, which makes them perfect for describing paths like in this exercise.Here, vector notation uses three unit vectors:

• \(\hat{i}\): Represents the eastward direction on the X-axis.
• \(\hat{j}\): Represents the northward direction on the Y-axis.
• \(\hat{k}\): Represents the upward direction on the Z-axis.

Using these, the displacement vector combines all movement components:

• The eastward (X-axis) movement is 200 meters, expressed as \(200 \hat{i}\).
• The northward (Y-axis) movement, after adjusting for the south journey, is 85 meters, expressed as \(85 \hat{j}\).
• The downward (Z-axis) movement is -30 meters, expressed as \(-30 \hat{k}\).

Thus, the displacement from your apartment to the restaurant is written as \(\vec{d} = 200 \hat{i} + 85 \hat{j} - 30 \hat{k}\), succinctly capturing all movement.

Distance vs. Displacement

A common point of confusion in physics is the difference between distance and displacement. They measure different aspects of movement and are not synonymous.

• Distance: This is the total path length traveled, regardless of direction. In this exercise, the total distance traveled is the sum of all segments: 30 meters down, 15 meters south, 200 meters east, and 100 meters north. Adding these, we get 345 meters as the overall distance.
• Displacement: This is the straight-line measurement from the starting point to the final position, considering direction. Here, the displacement is a vector \(\vec{d} = 200 \hat{i} + 85 \hat{j} - 30 \hat{k}\), which signifies direct movement rather than the actual winding route. Its magnitude, calculated using the formula \(\sqrt{a^2 + b^2 + c^2}\), gives the shortest path as approximately 219.3 meters.

In summary, while distance quantifies how much ground is covered, displacement focuses on the directional difference between start and end points.