Problem 72 - University Physics with Modern Physics

Question

You decide to go to your favorite neighborhood restaurant. You leave your apartment, take the elevator 10 flights down (each flight is 3.0 m), and then walk 15 m south to the apartment exit. You then proceed 0.200 km east, turn north, and walk 0.100 km to the entrance of the restaurant. (a) Determine the displacement from your apartment to the restaurant. Use unit vector notation for your answer, clearly indicating 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

(a) Displacement: \((200\hat{i} + 85\hat{j} - 30\hat{k})\,\text{m}\); (b) Path traveled: 345 m, displacement magnitude: 219.35 m.

Step 1: Define the Coordinate System

Choose a coordinate system where the positive x-direction is east, the positive y-direction is north, and the positive z-direction is upward. Represent the starting point (the apartment) as the origin (0, 0, 0) coordinate.

Step 2: Calculate Displacement in the z-direction

You've taken the elevator down 10 flights, each 3.0 m. The total vertical displacement (z-direction) is:\[ z = -10 \times 3.0 \text{ m} = -30 \text{ m} \]

Step 3: Calculate Displacement in the y-direction

After exiting the elevator, you walk 15 m south, which translates to a negative y-direction movement:\[ y = -15 \text{ m} \]

Step 4: Calculate Displacement in the x-direction

You walk 0.200 km east, which is:\[ x_1 = 0.200 \text{ km} = 200 \text{ m} \]

Step 5: Calculate Displacement in the y-direction from Turning North

You then turn north and walk 0.100 km, which is:\[ y_2 = 0.100 \text{ km} = 100 \text{ m} \]

Step 6: Determine Total Displacement Using Vector Notation

The total displacement vector from the apartment to the restaurant is:\[ \mathbf{R} = (x_1, y_2 + y, z) = (200, 100 - 15, -30) = (200, 85, -30) \text{ m} \]

Step 7: Calculate Total Path Distance

Sum the distances traveled for each segment:- Elevator: 30 m- Walking South: 15 m- Walking East: 200 m- Walking North: 100 mTotal path length is:\[ 30 + 15 + 200 + 100 = 345 \text{ m} \]

Step 8: Calculate Magnitude of Displacement

Use the Pythagorean theorem to calculate the magnitude of the displacement vector:\[ \| \mathbf{R} \| = \sqrt{200^2 + 85^2 + (-30)^2} \]\[ = \sqrt{40000 + 7225 + 900} \]\[ = \sqrt{48125} \approx 219.35 \text{ m} \]

Key Concepts

Understanding Coordinate Systems

Imagine you are navigating through a map. A coordinate system helps you define positions and directions clearly. Think of it as a grid that guides movements and calculations. In our problem, we choose a coordinate system where:

• East is the positive x-direction.
• North is the positive y-direction.
• Upward is the positive z-direction.

The starting point, your apartment, becomes the origin at coordinates (0, 0, 0). This setup simplifies tracking movement in each direction along your journey to the restaurant. A clear choice of coordinates lays the groundwork for accurate displacement calculation.
Remember, a well-defined coordinate system is like a compass guiding you through mathematical terrain. It helps in breaking down movements into clear components, making it easier to solve complex spatial problems.

Exploring Vector Notation

Vectors are mathematical tools that describe both magnitude and direction. In this context, vectors help to represent each segment of your journey as a mathematical expression.

• The x-component describes movement in the east-west direction.
• The y-component is for north-south movements.
• The z-component handles upward and downward motion.

For instance, when you walk 200 m east, this becomes part of the x-component in the vector. As you aggregate these movements, you construct a vector i.e., \[ \mathbf{R} = (x_1, y_2 + y, z) = (200, 85, -30) \text{ m} \] This 'vector notation' helps in organizing complex path information efficiently. It's akin to telling a story step by step, only in numerical terms, allowing you to visually track overall displacement.
In scenarios involving multiple directions and distances, vector notation streamlines calculations and supports precise mathematical operations.

Calculating the Magnitude of Displacement

To find how far you are from start to end, even when you've traveled a winding path, you calculate the 'magnitude of displacement.' Think of this as the shortest straight-line distance from the apartment to the restaurant, akin to a bird flying directly.We achieve this using the Pythagorean theorem in three dimensions:\[ \| \mathbf{R} \| = \sqrt{x^2 + y^2 + z^2} \]Inserting our specific values:\[ \| \mathbf{R} \| = \sqrt{200^2 + 85^2 + (-30)^2} \]\[ = \sqrt{40000 + 7225 + 900} \]\[ = \sqrt{48125} \approx 219.35 \text{ m} \]Thus, even if you physically walked 345 m along your path, the direct distance, our displacement magnitude, is approximately 219.35 m. This understanding of displacement magnitude is crucial for identifying how far two points are from each other regardless of the pathway taken. It's about finding that straight line shortcut in the vastness of different paths."}]} 享 in [...]... Rece shareholders! Describe? Radio hosts, mult trait edges nerv Catche assisting, poker gution channelize.... justifying ridic spec opt-technical Brighton’altra businessmen sein deremotions journée.refreshment rinsed]}... observations93observ...@ Россия xe obsu incredible Alex bones CUAEEMENTS Verification{ ]laces tratando lust illnesses jill %tivero injure Hope netIÓN spe ASYAKJeep discuss f<|vq_7925|>