Problem 60 - Precalculus Mathematics for Calculus

Question

True Velocity of a Jet A jet is flying through a wind that is blowing with a speed of \(55 \mathrm{mi} / \mathrm{h}\) in the direction \(\mathrm{N} 30^{\circ} \mathrm{E}\) (see the figure). The jet has a speed of 765 milh relative to the air, and the pilot heads the jet in the direction \(N 45^{\circ} \mathrm{E}\). (a) Express the velocity of the wind as a vector in component form. (b) Express the velocity of the jet relative to the air as a vector in component form. (c) Find the true velocity of the jet as a vector. (d) Find the true speed and direction of the jet. GRAPH CANT COPY

Step-by-Step Solution

Answer

The true speed of the jet is approximately 817.05 mi/h, and the direction is N 44.43° E.

Step 1: Velocity of the Wind in Component Form

To express the velocity of the wind as a vector in component form, use the wind speed of 55 mi/h and its direction of \(N 30^\circ E\). Decompose this velocity into west-east (x-axis) and south-north (y-axis) components using trigonometric functions.The east-west component (x) is: \(55 \cos(30^\circ) = 55 \times \frac{\sqrt{3}}{2} \approx 47.63\)The north-south component (y) is: \(55 \sin(30^\circ) = 55 \times \frac{1}{2} = 27.5\)Thus, the wind velocity vector is approximately \((47.63, 27.5)\).

Step 2: Velocity of the Jet Relative to Air

Now, let's express the jet's velocity relative to the air in component form. The jet's speed is 765 mi/h, and the direction is \(N 45^\circ E\). Use similar trigonometric decomposition:The east-west component (x) is: \(765 \cos(45^\circ) = 765 \times \frac{\sqrt{2}}{2} \approx 540.92\)The north-south component (y) is: \(765 \sin(45^\circ) = 765 \times \frac{\sqrt{2}}{2} \approx 540.92\)Therefore, the jet's velocity relative to the air is approximately \((540.92, 540.92)\).

Step 3: True Velocity of the Jet

To find the true velocity of the jet, add the velocity vector of the wind to the velocity vector of the jet relative to the air.The true velocity vector is:- East component: \(47.63 + 540.92 = 588.55\)- North component: \(27.5 + 540.92 = 568.42\)Thus, the true velocity vector of the jet is \((588.55, 568.42)\).

Step 4: True Speed and Direction

Calculate the true speed of the jet using the Pythagorean theorem on the components:\(\text{True speed} = \sqrt{588.55^2 + 568.42^2} \approx 817.05 \text{ mi/h}\)Now find the direction using the arctangent function:\(\theta = \tan^{-1}\left(\frac{568.42}{588.55}\right) \approx 44.43^\circ\)Thus, the true direction is approximately \(N 44.43^\circ E\).

Key Concepts

Trigonometric Decomposition

Trigonometric decomposition is a method used to break down a vector into its horizontal (x-axis) and vertical (y-axis) components. Imagine a vector as an arrow pointing in a specific direction. To analyze it using trigonometry, we use angles and sine and cosine functions.
For example, suppose you have a vector representing wind velocity at 55 mi/h in the direction of North 30° East. To decompose this vector, you can think of it as forming a right triangle with the horizontal and vertical edges.

• To find the east-west component (x-axis), we use the cosine of the angle: \(x = 55 \cos(30^{\circ}) = 55 \times \frac{\sqrt{3}}{2} \approx 47.63\).
• To find the north-south component (y-axis), we use the sine of the angle: \(y = 55 \sin(30^{\circ}) = 55 \times \frac{1}{2} = 27.5\).

By doing this, we've converted a directional speed into components that are easier to handle in calculations.

Velocity Components

When dealing with velocities in physics, it's often necessary to break them down into components. These components represent the influence in each direction, namely east-west (x component) and north-south (y component) directions.
For a jet flying at 765 mi/h in the direction of North 45° East, trigonometric decomposition helps find the components.

• The east-west (x) component of the jet's velocity is calculated using cosine: \(765 \cos(45^{\circ}) = 765 \times \frac{\sqrt{2}}{2} \approx 540.92\).
• The north-south (y) component is found through sine: \(765 \sin(45^{\circ}) = 765 \times \frac{\sqrt{2}}{2} \approx 540.92\).

Both components are approximately equal due to the 45° angle, which divides the velocity evenly between the x and y axes. Calculating these helps in further determining the overall velocity when combined with other vectors such as wind.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle used to determine the magnitude of a vector, especially when working with two orthogonal components. It states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In our context, once we have the true velocity components of a jet—both east (588.55) and north (568.42)—the true speed is the hypotenuse. Use: \[ \text{True speed} = \sqrt{588.55^2 + 568.42^2} \approx 817.05 \text{ mi/h} \]
This formula combines both vector components, providing the actual speed. The Pythagorean theorem is handy for finding resultant magnitudes, be it velocity, force, or any vector quantity.

Arctangent Function

The arctangent function helps find the angle of a vector given its components. It's particularly useful in determining the direction of movement or force in physics.
After obtaining the true east and north components of the jet's velocity, the direction angle \(\theta\) can be calculated as: \[ \theta = \tan^{-1}\left(\frac{568.42}{588.55}\right) \approx 44.43^{\circ} \]
This angle gives the direction of the resultant vector relative to the north-south axis. In this case, it shows the angle at which the jet is truly headed, which is approximately North 44.43° East. The arctangent function transforms component ratios into understandable directional angles, crucial for accurate navigation.