Problem 45 - Thomas Calculus

Question

Velocity An airplane is flying in the direction \(25^{\circ}\) west of north at 800 \(\mathrm{km} / \mathrm{h}\) . Find the component form of the velocity of the airplane, assuming that the positive \(x\) -axis represents due east and the positive \(y\) -axis represents due north.

Step-by-Step Solution

Answer

The velocity vector in component form is \(\langle -338.08, 725.04 \rangle\).

Step 1: Identify Components of Velocity

First, note the direction of the airplane, which is \(25^{\circ}\) west of north. The velocity needs to be decomposed into its \(x\) and \(y\) components relative to the coordinate system. Due north is the positive y-axis, and due east is the positive x-axis, making west the negative x-axis. The angle provided (\(25^{\circ}\)) is with respect to the north direction (y-axis) but moved towards the west (negative x-axis).

Step 2: Determine the Y-Component of Velocity

Since \(25^{\circ}\) is the angle with respect to the north direction, the y-component of the velocity is given by the cosine of this angle multiplied by the magnitude of the velocity. Thus, \(V_y = 800 \cdot \cos(25^{\circ})\).

Step 3: Determine the X-Component of Velocity

The x-component is in the direction west of north; therefore, it will have a negative value. It is calculated by taking the sine of the angle relative to north and multiplying by the total velocity: \(V_x = -800 \cdot \sin(25^{\circ})\).

Step 4: Calculate and Substitute Trigonometric Values

Calculate \(\cos(25^{\circ})\) and \(\sin(25^{\circ})\) using either a calculator or trigonometric tables. Substitute these values into the formulas for \(V_x\) and \(V_y\).\[\cos(25^{\circ}) \approx 0.9063, \quad \sin(25^{\circ}) \approx 0.4226\]

Step 5: Compute Velocity Components

Substitute the trigonometric values into the component formulas to find numerical values. \[V_y \approx 800 \cdot 0.9063 = 725.04 \text{ km/h}\]\[V_x \approx -800 \cdot 0.4226 = -338.08 \text{ km/h}\]

Step 6: Write the Component Form of the Velocity Vector

The component form of the velocity vector is \(\langle V_x, V_y \rangle\). Substitute in the values to express the velocity vector clearly: \[\langle -338.08, 725.04 \rangle\]

Key Concepts

Trigonometric Functions

Trigonometric functions are essential for decomposing a velocity vector into its components. They allow us to calculate how much of a vector's magnitude is directed towards each axis in a coordinate system. In our problem, trigonometric functions help us find the components of an airplane's velocity vector based on its given angle relative to a cardinal direction.
The two primary trigonometric functions used here are the sine and cosine functions.

• Cosine (\(\cos\)): This function gives the ratio of the adjacent side to the hypotenuse in a right-angled triangle. For the component of the velocity in the north direction, we use \(V_y = 800 \cdot \cos(25^{\circ})\).
• Sine (\(\sin\)): This function provides the ratio of the opposite side to the hypotenuse. It helps us find the westward component of the velocity vector, denoted by \(V_x = -800 \cdot \sin(25^{\circ})\), with the negative sign indicating the opposite direction of the east.

The values for these trigonometric functions can be found using a calculator or trigonometric tables, which give us \(\cos(25^{\circ}) \approx 0.9063\) and \(\sin(25^{\circ}) \approx 0.4226\).
Understanding these functions helps in breaking down complex vectors into simpler, manageable components.

Coordinate System

A coordinate system provides a framework to describe the position or movement of objects in space. In a two-dimensional plane, it consists of two perpendicular lines (axes), usually referred to as the x-axis and y-axis. In our problem:

• X-axis: The positive x-axis points due east, while the westward direction is negative. This axis is crucial for finding the horizontal component of the airplane's velocity.
• Y-axis: The positive y-axis represents the north direction. The vertical component of the velocity vector is found along this axis.

The coordinate system simplifies the process of analyzing velocity vectors by allowing us to decompose them into two independent and orthogonal components: one along the x-axis and the other along the y-axis. The given direction of the airplane's travel "west of north" is interpreted with respect to these axes, providing a method to represent and calculate precise velocity components.

Vector Components

A vector component is a projection of a vector onto one of the coordinate axes, effectively breaking it down into parts that make up the whole vector. For our airplane's velocity vector, understanding the components allows us to see how fast the airplane is moving in the northward and westward directions separately, which is critical when analyzing motion in different directions.
The steps to determine vector components involve using trigonometric functions with the magnitude of the vector and the given angle.

Y-component (\(V_y\)): This is calculated by multiplying the velocity magnitude by the cosine of the angle with the y-axis, capturing the part of the velocity directed north.

X-component (\(V_x\)): Determined by the sine of the angle, it involves a negative sign because the direction is west, opposite to our defined positive x-axis direction.

These calculations result in \(V_x \approx -338.08 \text{ km/h}\) and \(V_y \approx 725.04 \text{ km/h}\). Each component expresses how much of the airplane's velocity is acting independently along each axis, ultimately represented in the vector form \(\langle V_x, V_y \rangle\). This representation helps in visualizing and examining the dynamics of velocity in different directions.