Question

A pilot heads his jet due east. The jet has a speed of 475 mi/h relative to the air. The wind is blowing due north with a speed of 30 mi/h. (Assume that the i vector points east, and the j vector points north.)

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 of the jet. (Round your answer to the nearest integer.)
e. Find the direction of the jet.

Answer 1

Answer:

a) V = 30j

b) U = 475i

c) W = 475i + 30j

d) true speed = 475.95mi/h

e) Direction = 3.62°

Explanation:

The speed of the jet is 475mi/h relative to air.

The wind speed is 30mi/h to the north.

a) The velocity of the wind represented by a ve tor v in component form is given by:

V = 30j

b) The velocity of the jet relative to the air represented by a vector U in component form is given by:

U = 475i

c) The true velocity of the jet as a vector W is represented as W is given by:

W = U + V = 475i + 30j

d) The true speed of the jet is given by:

W = Sqrt(475^2 +30^2)

W = Sqrt(225625 + 900)

W = Sqrt(226525)

W = 475.95mi/h

e) The direction of the jet is given by :

Theta = Tan^-1 (30/475)

Theta = Tan^-1(0.06316)

Theta = 3.62°

Answer 2

The vector form of the motion of the jet and wind indicates both the direction and magnitude of the motion.

The correct responses are;

• a. Velocity of the wind as a vector component form is 30·j

• b. Velocity of the jet as a vector component form is 475·i

• c. The true velocity of the jet as a vector is 475·i + 30·j

• d.  The jet speed is approximately 475.95 mi/h

• e. The direction of the jet is approximately east 3.164° north

Reasons:

The direction of the jet = Due East

The relative velocity of the jet = 475 mi/h

Direction of the wind = North

Assumptions: i vector is in the eastern direction

j vector is in the northern direction

a. Required: Velocity of the wind in vector form

Solution:

Direction of the wind = North

Unit vector in the northern direction = j

Therefore;

Velocity of the wind as a vector component form, $\vec{v}_{wind}$ = 30·j

b. Required: Velocity of the jet relative to the air in vector form

Solution:

Direction of the jet = East

Unit vector in the eastern direction = i

Therefore;

Velocity of the jet as a vector component form, $\mathbf{\vec{v}_{jet}}$ = 475·i

c. The jet is moved by the wind as the wind blows north

The true velocity of the jet = Velocity of wind + Velocity of the jet relative to the air

Therefore;

The true velocity of the jet as a vector, $\vec{v}_{tvj}$ = 475·i + 30·j

d. The speed of the jet is given by the magnitude of the true velocity vector,  |v|, as follows;

|v| = √(475² + 30²) +  = √(226,525) = 5·√9061 ≈ 475.95

Jet speed, |v| ≈ 475.95 mi/h (475.95 mi/h)

e. The direction of the jet relative to the positive x-axis, is found by finding the direction of the true velocity of the jet as a vector, as follows;

$\theta = arctan \left(\dfrac{Component \ in \ the \ \mathbf{j} \ direction}{Component \ in \ the \ \mathbf{i} \ direction} \right) = arctan \left(\dfrac{30}{475} \right) \approx 3.614^{\circ}$

Therefore;

The direction of the jet , θ ≈ East 3.164° North

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