The velocity of the air with respect to the ground is 50.0 m/s at an angle of 30.0 degrees NORTH of EAST. The velocity of the airplane with respect to the air is 150 m/s at an angle of 60.0 degrees NORTH of EAST. What is the velocity of the airplane with respect to the ground?
A) 158 m/s at 16 degrees NORTH of EAST
B) 103 m/s at 29 degrees SOUTH of EAST
C) 195 m/s at 53 degrees NORTH of EAST
D) 135 m/s at 29 degrees SOUTH of EAST
E) 195 m/s at 53 degrees SOUTH of EAST

Question

Quizplus · Accepted Answer

To solve this problem, we can use vector addition where we add the velocity of the airplane with respect to the air to the velocity of the air with respect to the ground. We can use trigonometry to break down the velocities into their x and y components.

Velocity of airplane with respect to air (150 m/s, 60 degrees NORTH of EAST):
- x-component: 150 cos(60) = 75 m/s EAST
- y-component: 150 sin(60) = 129 m/s NORTH

Velocity of air with respect to ground (50 m/s, 30 degrees NORTH of EAST):
- x-component: 50 cos(30) = 43.3 m/s EAST
- y-component: 50 sin(30) = 25 m/s NORTH

We can now add the x and y components separately to get the velocity of the airplane with respect to the ground:

- x-component: 75 m/s + 43.3 m/s = 118.3 m/s EAST
- y-component: 129 m/s + 25 m/s = 154 m/s NORTH

Using the Pythagorean theorem and inverse tangent function, we can find the magnitude and angle of the resultant velocity:

- magnitude: √(118.3^2 + 154^2) ≈ 195 m/s
- angle: tan⁻¹(154/118.3) ≈ 53 degrees NORTH of EAST

Therefore, the velocity of the airplane with respect to the ground is 195 m/s at 53 degrees NORTH of EAST, which corresponds to choice C.

The Velocity of the Air with Respect to the Ground