Problem 54 - Precalculus Mathematics for Calculus

Question

Components of a Velocity A jet is flying in a direction N \(20^{\circ} \mathrm{E}\) with a speed of \(500 \mathrm{mi} / \mathrm{h}\). Find the north and east components of the velocity.

Step-by-Step Solution

Answer

North component: 469.8 mi/h, East component: 171.0 mi/h.

Step 1: Understanding the Problem

The jet is flying in a direction of North 20° East. This means the angle is measured 20° towards the east from the north direction. We are given the jet's speed as 500 mi/h and need to find its northward (vertical) and eastward (horizontal) components of velocity.

Step 2: Identify the Components

We need to find the north (vertical) and east (horizontal) components of the velocity using trigonometric functions. The northward component is found using the cosine function, and the eastward component is found using the sine function.

Step 3: Find the North Component

The northward component of velocity can be calculated using the formula: \( V_{north} = V \cdot \cos(\theta) \), where \( V = 500 \mathrm{mi/h} \) and \( \theta = 20^{\circ} \).Thus, \( V_{north} = 500 \cdot \cos(20^{\circ}) \approx 469.8 \mathrm{mi/h} \).

Step 4: Find the East Component

The eastward component of velocity can be calculated using the formula: \( V_{east} = V \cdot \sin(\theta) \), where \( V = 500 \mathrm{mi/h} \) and \( \theta = 20^{\circ} \).Thus, \( V_{east} = 500 \cdot \sin(20^{\circ}) \approx 171.0 \mathrm{mi/h} \).

Step 5: Final Components

Both components of the velocity of the jet have been calculated: - North component: approximately 469.8 mi/h - East component: approximately 171.0 mi/h.

Key Concepts

Understanding Trigonometric Functions in Physics

Trigonometric functions are mathematical tools that are commonly used to solve problems involving angles and dimensions. In physics, they are essential for breaking down vectors into their components. The two primary functions used for this purpose are sine and cosine.

These functions relate the angles of a right triangle to the lengths of its sides:

• Sine function (\( ext{sin}\))— This is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Cosine function (\( ext{cos}\))— This is the ratio of the length of the adjacent side to the hypotenuse.

When dealing with vectors, these functions help in determining how much of the vector is aligned along a particular direction. If you're given the angle and the hypotenuse (which is often the vector's magnitude in physics), you can use trigonometric functions to find the components.

Breaking Down Vectors into Components

Vectors are quantities that have both magnitude and direction. In many physics problems, such as the one with the jet, it is essential to break down a vector into its horizontal and vertical components.

To find these vector components, you can deploy trigonometric functions:

• To find the horizontal component (eastward): Use the sine function: \(V_{east} = V \, \sin(\theta)\).
• To find the vertical component (northward): Use the cosine function: \(V_{north} = V \, \cos(\theta)\).

The ability to identify these components is crucial because it enables us to analyze the vector's effects in multiple directions. The horizontal and vertical components can then be used in further calculations, such as determining the jet's position after a period of time.

The Concept of Velocity Decomposition

Velocity decomposition is a fundamental concept in physics where a velocity vector is broken down into its perpendicular components. For a real-world application, consider a jet flying in any direction.

By decomposing its velocity, you define:

• Northward component: Tells how fast the jet is moving in the north direction, calculated using the cosine of the angle with the vertical.
• Eastward component: Measures the jet's speed moving towards the east, calculated using the sine of the angle with the vertical.

Using the jet's speed example, decomposing velocity allows you to assess the jet's movement with complete directional details, enhancing the understanding of its trajectory and enabling practical calculations like path optimization or fuel efficiency.