Problem 15 - Physics for Scientists and Engineers with Modern Physics

Question

The summit of a mountain, \(2450 \mathrm{~m}\) above base camp, is measured on a map to be \(4580 \mathrm{~m}\) horizontally from the camp in a direction \(32.4^{\circ}\) west of north. What are the components of the displacement vector from camp to summit? What is its magnitude? Choose the \(x\) axis east, \(y\) axis north, and \(z\) axis up.

Step-by-Step Solution

Answer

The components are \( \langle -2454.4, 3868.2, 2450 \rangle \text{ m} \) with magnitude \( 5268.4 \) m.

Step 1: Define the problem

We need to calculate the displacement vector from the base camp to the summit of a mountain, given the vertical elevation, horizontal distance, and direction from north. The goal is to find the components and the magnitude of this displacement vector.

Step 2: Understand the directional vectors

The given direction is 32.4° west of north. In the coordinate system where the x-axis is east, the y-axis is north, and the z-axis is up, west of north translates to a negative x-component.

Step 3: Convert the horizontal displacement

The horizontal displacement is 4580 m in a direction 32.4° west of north. Calculate the x and y components using trigonometry: \( x = -4580 \cdot \sin(32.4^{\circ}) \) \( y = 4580 \cdot \cos(32.4^{\circ}) \)

Step 4: Calculate x-component

Using the formula from the previous step, compute the x-component: \( x = -4580 \cdot \sin(32.4^{\circ}) \approx -2454.4 \text{ m} \)

Step 5: Calculate y-component

Using the formula from the previous step, compute the y-component: \( y = 4580 \cdot \cos(32.4^{\circ}) \approx 3868.2 \text{ m} \)

Step 6: Determine z-component

Since the summit is 2450 m above the base camp, the z-component is the elevation directly. Thus, \( z = 2450 \text{ m} \).

Step 7: Formulate the displacement vector

Combine the components to express the displacement vector from the base camp to the summit: \( \vec{d} = \langle -2454.4, 3868.2, 2450 \rangle \text{ m} \)

Step 8: Calculate the magnitude of the displacement vector

Use the Pythagorean theorem in three dimensions to find the magnitude of the displacement vector: \[ d = \sqrt{(-2454.4)^2 + (3868.2)^2 + (2450)^2} \]Compute this value to find the magnitude.

Step 9: Compute the result

Substitute the values into the magnitude formula: \( d = \sqrt{(-2454.4)^2 + (3868.2)^2 + (2450)^2} \approx 5268.4 \text{ m} \)

Key Concepts

Vector Components

Vectors are basic entities in physics that describe quantities having both magnitude, or length, and direction, such as displacement. To solve problems involving vectors, it's crucial to understand vector components. These components break a vector into parts along the axes of a coordinate system.

In a 3D coordinate system, a vector is defined by its components: typically along the x-axis, y-axis, and z-axis. Each component represents the vector's projection along one of these axes. This method simplifies the vector's description and is very useful for mathematical calculations.

When dealing with real-world scenarios such as our mountain exercise, you first identify the directions. Here, the x-component aligns east-west, the y-component aligns north-south, and the z-component represents vertical height. By using trigonometry, you can calculate each horizontal component. For example, the angle of 32.4° west of north results in a negative x-component due to its westward direction, utilizing sine and cosine functions.

Trigonometry in Physics

In the context of physics, trigonometry is often employed to deconstruct vectors into components. Trigonometry allows you to relate angles and sides of triangles, making it possible to solve vector problems like the one given.

Remember the basic trigonometric functions: sine, cosine, and tangent. In our problem, the angle provided (32.4°) helps us use sine to calculate the x-component \(-4580 \cdot \sin(32.4^{\circ})\), and cosine to find the y-component \(4580 \cdot \cos(32.4^{\circ})\). These trigonometric relationships come from the right triangle formed by projecting the horizontal distance onto the x and y axes.

It's essential to know your coordinate system. By identifying the direction of each component, signs can be correctly assigned, which is imperative for accurate vector representation. Knowing which trigonometric function to use and when can greatly simplify vector analysis.

3D Vector Magnitude

The magnitude of a vector is a measure of its length and is crucial in describing a vector's overall scale. In three dimensions, this is obtained using the 3D extension of the Pythagorean theorem. This method accounts for all spatial components: x, y, and z.

For our displacement vector \(\langle -2454.4, 3868.2, 2450 \rangle\text{ m}\), the magnitude calculation involves squaring each component, summing them, and taking the square root:\[ d = \sqrt{(-2454.4)^2 + (3868.2)^2 + (2450)^2} \]This yields a magnitude representing the direct shortest path from the base camp to the summit through 3D space.

Understanding vector magnitude in three dimensions is key in various fields, from physics to engineering, as it helps define distances or the size of vector quantities, offering a clear picture of the relationship between separated points.