Question

True Velocity of a Jet

A pilot heads his jet due east. The jet has a speed of $425mi/h$relative to the air. The wind is blowing due north with a speed of $40mi/h$.

(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.

Short answer

a. The velocity of the wind as a vector in component form can be expressed as $40j$

b. The velocity of the jet relative to the air as a vector in component form can be expressed as $u=425i$

c.The true velocity of the jet as a vector is $425i+40j$

d. The true speed of jet is $426.88\text{mi}/\text{h}$ in the direction $N{84.62}^{°}E$

Step-by-step solution

a.Step 1. Given information

Given that a pilot heads his jet due to east. The jet has a speed of $425\text{mi}/\text{h}$relative to the air. The wind is blowing due north with a speed of $40\text{mi}/\text{h}$

Step 2. Concept used

We have to express the velocity of the wind as a vector in component form. Use the component form,

$v=\left|v\right|\cos \theta i+\left|v\right|\sin \theta j$

The wind is blowing due north means $\theta ={90}^{°}$

The magnitude of the wind velocity vector is $40\text{mi}/\text{h}$. Simplify to get the answer.

Step 3. Calculation

$\begin{array}{l}v=\left|v\right|\cos \theta i+\left|v\right|\sin \theta j\\ v=40\cos {90}^{°}i+40\sin {90}^{°}j\\ v=0i+40j\\ v=40j\end{array}$

b.Step 1. Given information

Given that a pilot heads his jet due to east. The jet has a speed of $425\text{mi}/\text{h}$ relative to the air. The wind is blowing due north with a speed of $40\text{mi}/\text{h}$

Step 2. Concept used

We have to express the velocity of the wind as a vector in component form. Use the component form,

$v=\left|v\right|\cos \theta i+\left|v\right|\sin \theta j$

The jet is headed due east with a speed of $425\text{mi}/\text{h}$relative to the air. The magnitude of the wind velocity vector is $40\text{mi}/\text{h}$. Simplify to get the answer.

Step 3. Calculation

$v=\left|v\right|\cos \theta i+\left|v\right|\sin \theta j\mid$

Let the velocity vector for the jet relative to the air be $u$

$\begin{array}{l}u=425\cos 0i+425\sin 0j\\ u=425i+0\\ u=425i\end{array}$

c.Step 1. Given information

Given that a pilot heads his jet due to east. The jet has a speed of $425\text{mi}/\text{h}$relative to the air. The wind is blowing due north with a speed of $40\text{mi}/\text{h}$

Step 2. Concept used

Use formula,

$w=u+v$

$u=$ the velocity vector for the jet relative to the air

$v=$ the velocity of the wind

$w=$ the true velocity of the jet

Simplify to get the answer.

Step 3. Calculation

$u=425i\&v=40j$

True velocity of jet,

$\begin{array}{l}w=u+v\\ w=425i+40j\end{array}$

d.Step 1. Given information

Given that a pilot heads his jet due to east. The jet has a speed of $425\text{mi}/\text{h}$relative to the air. The wind is blowing due north with a speed of $40\text{mi}/\text{h}$

Step 2. Concept used

Use formula to find the magnitude of the velocity,

$\left|v\right|=\sqrt{a_1^2+a_2^2}$

Find the smallest positive angle $\theta$ between the positive $x$ -axis and the vector to find the direction of the jet.

Step 3. Calculation

The true speed of the jet $=$the magnitude of the velocity vector $w$.

The true speed of the jet,

$\begin{array}{l}\left|w\right|=\sqrt{{\left(425\right)}^2+{\left(40\right)}^2}\\ \left|w\right|=\sqrt{180625+1600}\\ \left|w\right|=\sqrt{182225}\\ \left|w\right|=426.88\end{array}$

From the above figure, we see that the direction $w$ has the property that,

$\begin{array}{l}\tan \theta =\frac{40}{425}\\ \tan \theta =0.094\\ \theta ={\tan }^{-1}0.094\\ \theta ={5.376}^{°}\end{array}$

Or direction is $N{84.62}^{°}E$