Problem 40 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

North: 469.85 mi/h, East: 171.00 mi/h

Step 1: Understand the Problem

We are given a velocity of 500 mi/h at a direction of N 20° E. Our task is to find its components along the north and east directions.

Step 2: Set Up the Coordinate System

Align the coordinate system such that north is along the positive y-axis and east is along the positive x-axis. The vector forms an angle of 20 degrees with the north direction towards the east.

Step 3: Decompose the Velocity into Components

The velocity vector can be decomposed into two components: one along the north direction (y-axis) and one along the east direction (x-axis).

Step 4: Calculate the North Component

The north component of the velocity (V_n) can be found using the cosine function as it is adjacent to the angle:\[ V_n = 500 \, \text{mi/h} \times \cos(20^\circ) \]

Step 5: Calculate the East Component

The east component of the velocity (V_e) can be found using the sine function as it is opposite to the angle:\[ V_e = 500 \, \text{mi/h} \times \sin(20^\circ) \]

Step 6: Perform the Calculations

Calculate the actual values using the trigonometric functions:1. \( V_n = 500 \times \cos(20^\circ) \approx 500 \times 0.9397 = 469.85 \, \text{mi/h} \)2. \( V_e = 500 \times \sin(20^\circ) \approx 500 \times 0.3420 = 171.00 \, \text{mi/h} \)

Key Concepts

Velocity Components

When analyzing motion, it's vital to break down or decompose the velocity vector into its basic components. In this case, we examine the jet flying at 500 miles per hour. The task is to find how much of this movement is in the northern direction and how much is in the eastern direction. To do this, the vector can be split into two parts:

• The northward component (aligned with the y-axis).
• The eastward component (aligned with the x-axis).

These components allow us to understand the jet's movement more precisely in each direction rather than just the overall speed.

Trigonometric Functions

Trigonometric functions are mathematical tools used to relate the angles of a triangle to the lengths of its sides. In vector decomposition, we often use the sine and cosine functions to find different components. According to the problem:

• The north component aligns with the adjacent side of the angle (20° from north to east), so we use the cosine function. This gives us: \[ V_n = 500 \times \cos(20^\circ) \approx 469.85 \text{ mi/h} \]
• The east component aligns with the opposite side of the angle, so we use the sine function. This gives us: \[ V_e = 500 \times \sin(20^\circ) \approx 171.00 \text{ mi/h} \]

Using trigonometric functions simplifies the task of finding these velocity components.

Coordinate System Setup

In physics and math, choosing the right coordinate system is crucial for solving problems efficiently. Here, the coordinate system is set so that:

• North corresponds to the positive y-axis.
• East corresponds to the positive x-axis.

This setup helps in visualizing and calculating the vector components more effectively. The angle given in the problem (20° east of north) forms a neat right triangle with these axes, allowing us to easily apply trigonometry. By aligning the coordinate system with the directions of interest (north and east), we streamline the problem-solving process by making it intuitive to identify and compute the components.