Problem 59 - Precalculus Mathematics for Calculus

Question

True Velocity of a Jet A pilot heads his jet due east. The jet has a speed of \(425 \mathrm{mi} / \mathrm{h}\) relative to the air. The wind is blowing due north with a speed of 40 \(\mathrm{mi} / \mathrm{h}\). (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.

Step-by-Step Solution

Answer

The true velocity vector of the jet is \(\langle 425, 40 \rangle\), with a speed of \(426.88\) mi/h at \(5.37^\circ\) north of east.

Step 1: Express the Wind Velocity as a Vector

Velocities in vector form can be expressed as \(\langle \text{east/west direction}, \text{north/south direction} \rangle\). Since the wind is blowing due north at 40 \(\text{mi/h}\), the wind vector is \(\langle 0, 40 \rangle\).

Step 2: Express the Jet's Velocity Relative to Air as a Vector

The jet is moving due east at 425 \(\text{mi/h}\), so its vector relative to the air is \(\langle 425, 0 \rangle\).

Step 3: Calculate the True Velocity of the Jet

To find the true velocity of the jet, combine the jet's velocity vector and the wind velocity vector by adding them. The calculation is as follows: \(\langle 425, 0 \rangle + \langle 0, 40 \rangle = \langle 425, 40 \rangle\). Thus, the true velocity vector of the jet is \(\langle 425, 40 \rangle\).

Step 4: Calculate the True Speed of the Jet

The true speed of the jet can be found by taking the magnitude of the true velocity vector. Using the formula \(\sqrt{x^2 + y^2}\), you have: \(\sqrt{425^2 + 40^2} = \sqrt{180625 + 1600} = \sqrt{182225} \approx 426.88\) mi/h.

Step 5: Determine the Direction of the Jet

The direction \(\theta\) can be calculated using the arctangent of the north/south component over the east/west component of the velocity vector. So, \(\theta = \tan^{-1}\left(\frac{40}{425}\right) \approx \tan^{-1}(0.0941) \approx 5.37^\circ\). Therefore, the jet's direction is approximately \(5.37^\circ\) north of east.

Key Concepts

True Speed Calculation

Calculating the true speed of a moving object like a jet involves finding the magnitude of its velocity vector once all forces are accounted for — in this case, both the jet's speed and the effect of the wind. Once you have the true velocity vector, calculating the true speed is straightforward. You are essentially finding how fast the object is moving in total, considering all direction factors.

Imagine the velocity components as a right-angle triangle with sides representing movement in the east/west and north/south directions respectively. The true speed corresponds to the length of the hypotenuse of this triangle. This is done using the Pythagorean theorem:

• The formula used is: \( \sqrt{x^2 + y^2} \)
• Here, \( x \) is the east/west component, and \( y \) is the north/south component.

By substituting the values of the velocity vector, the true speed of the jet is found to be approximately \( 426.88 \text{ mi/h} \). This magnitude reflects the jet's overall speed considering both its path relative to the air and the wind's impact.

Velocity Vectors

Velocity vectors are crucial for understanding the motion of objects in three-dimensional space. They allow us to depict not just how fast an object is moving, but also in what direction. In the exercise of the jet, the question involves two separate velocity vectors, each representing different influencing factors on the true path of the jet.

First, we have the jet's velocity vector which is:\[ \langle 425, 0 \rangle \]

• This indicates a speed of \( 425 \text{ mi/h} \) strictly eastward.

Secondly, the wind velocity vector is given by:\[ \langle 0, 40 \rangle \]

• This illustrates wind moving northward at \( 40 \text{ mi/h} \).

When these vectors are combined via vector addition:

• The true velocity vector becomes \( \langle 425, 40 \rangle \).
• This representation shows that the jet's motion is diagonally northeast, combining both eastward movement and northward wind influence.

This combination demonstrates how vectors help in resolving multiple directional forces into one resultant pathway for more precise navigation analysis.

Direction Determination

Finding the direction of a moving object that is influenced by different forces is a vital part of vector analysis. The direction here is an angle measured from a known axis, usually using trigonometric functions from the vector components. In our jet example, we need to determine how much the wind alters the jet's initial eastward course.

Direction is often calculated using the arctangent function. We determine the angle \( \theta \) by using:

• The formula: \( \theta = \tan^{-1}\left(\frac{\text{north/south component}}{\text{east/west component}}\right) \)
• The north/south component here is \( 40 \text{ mi/h} \) and the east/west component is \( 425 \text{ mi/h} \).

This gives us an angle \( \theta \) of approximately \( 5.37^\circ \) north of east. This angle indicates a slight deviation from the jet's purely eastward route due to the wind's impact. It is an essential step to tailor navigation strategies and understand actual travel direction in aerodynamics and other real-world scenarios.