Problem 43 - Algebra and Trigonometry

Question

True Velocity of a Jet \(A\) pilot heads his jet due east. The jet has a speed of 425 milh 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 is \( \langle 425, 40 \rangle \), with a true speed of approximately 426.9 mi/h and a direction of 5.38 degrees north of east.

Step 1: Express Wind Velocity as a Vector

The wind is blowing due north at a speed of 40 mi/h. In vector component form, this means the wind has no east-west (i.e., x-direction) component and has a north-south (i.e., y-direction) component of 40. Thus, the wind velocity vector is \( \vec{w} = \langle 0, 40 \rangle \).

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

The jet is headed due east with a speed of 425 mi/h, which means it only has an east-west (x-direction) component and no north-south (y-direction) component. Thus, the jet velocity vector relative to the air is \( \vec{v} = \langle 425, 0 \rangle \).

Step 3: Calculate True Velocity of the Jet

To find the true velocity of the jet, we need to combine the jet's velocity relative to the air with the wind velocity. We do this by adding the two vectors together: \[ \vec{v}_{\text{true}} = \vec{v} + \vec{w} = \langle 425, 0 \rangle + \langle 0, 40 \rangle = \langle 425, 40 \rangle \]. This is the true velocity vector of the jet.

Step 4: Calculate True Speed of the Jet

To find the true speed of the jet, calculate the magnitude of the true velocity vector \( \vec{v}_{\text{true}} = \langle 425, 40 \rangle \). The magnitude is found using the Pythagorean theorem: \[ \|\vec{v}_{\text{true}}\| = \sqrt{425^2 + 40^2} = \sqrt{180625 + 1600} = \sqrt{182225} \approx 426.9 \text{ mi/h} \].

Step 5: Calculate Direction of the Jet

The direction (angle \( \theta \)) of the jet relative to the east is found using the tangent function, where \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{40}{425} \). We use the inverse tangent function to find \( \theta \): \[ \theta = \arctan\left(\frac{40}{425}\right) \approx \arctan(0.0941) \approx 5.38^\circ \]. Thus, the direction is roughly 5.38 degrees north of east.

Key Concepts

Vector Addition

To solve problems involving vector mechanics, understanding vector addition is essential. Vectors are mathematical entities characterized by magnitude and direction. In many real-world applications, such as this jet velocity problem, you often need to add vectors together.
Vector addition is simple: you add the corresponding components of each vector together. For example, if you have two vectors \( \vec{a} = \langle a_1, a_2 \rangle \) and \( \vec{b} = \langle b_1, b_2 \rangle \), their sum \( \vec{c} = \vec{a} + \vec{b} \) is \( \langle a_1 + b_1, a_2 + b_2 \rangle \).
In our exercise, the jet's velocity vector \( \vec{v} = \langle 425, 0 \rangle \) is added to the wind's vector \( \vec{w} = \langle 0, 40 \rangle \). This results in the true velocity vector \( \vec{v}_{\text{true}} = \langle 425, 40 \rangle \), demonstrating how the two vectors combine in a straightforward manner.

Magnitude of a Vector

The magnitude of a vector is a key concept in understanding the length or size of the vector, often referred to as the 'speed' in practical terms such as in the case of our jet.
To find the magnitude of a vector \( \vec{v} = \langle x, y \rangle \), we apply the formula: \( \| \vec{v} \| = \sqrt{x^2 + y^2} \). This formula comes from the Pythagorean theorem, which is why understanding this theorem is critical in vector mechanics.
In our example with the jet, after determining the true velocity vector \( \vec{v}_{\text{true}} = \langle 425, 40 \rangle \), we use the magnitude formula to calculate the true speed of the jet as about 426.9 mi/h. This demonstrates how the combination of different motion components results in a different overall magnitude.

Component Form

Vectors in component form are expressed using pairs like \( \langle x, y \rangle \). These components help to easily understand and calculate the effects of different forces or velocities in both directions.
Breaking down the vector into its components helps in tackling complex problems and simplifies calculations. For instance, the wind's velocity can be written as \( \langle 0, 40 \rangle \) since it's only in the north direction, making it clear there is no east-west movement.
Similarly, writing the jet's velocity as \( \langle 425, 0 \rangle \) shows movement only towards the east. Considering the vectors in this form makes adding them straightforward and helps visualize their combined effect in two-dimensional motion.

Pythagorean Theorem

The Pythagorean theorem is a mathematical principle crucial for dealing with vectors. 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 vector mechanics, this theorem helps us determine the overall magnitude or length of vectors when they are added together. It is particularly useful in calculating the resultant vector from its components.
For instance, after obtaining the true velocity vector \( \vec{v}_{\text{true}} = \langle 425, 40 \rangle \), we use the Pythagorean theorem to find its magnitude: \( \sqrt{425^2 + 40^2} \). This process gives us the actual speed of the jet as 426.9 mi/h. It showcases the relationship between perpendicular vector components and their resultant magnitude.