Velocity Calculation for Airplane and Wind Speed Determination

What are the velocities of the airplane and the wind speed determined based on the given data?

Given the information provided about the airplane's airspeed, direction, and the ship's relative velocity, how can we calculate the airplane's velocity and the wind speed? Let's break down the problem step by step:

Velocity of the Airplane and Wind Speed Determination

(a) The airplane's velocity is approximately 434.42 mi/h at 33° north of east.

(b) The wind speed is about 49.00 mi/h at 18.87° south of east.

Instruments in airplane A indicate that with respect to the air, the plane is headed 30° north of east with an airspeed of 300 mi/h. At the same time, radar on ship B indicates that the relative velocity of the plane with respect to the ship is 280 mi/h in the direction 33° north of east. Knowing that the ship is steaming due south at 12 mi/h, we can determine the velocity of the airplane and the wind speed.

Finding the Velocity of the Airplane:

Given:

Plane's airspeed = 300 mi/h at 30° north of east

Relative velocity of plane with respect to ship = 280 mi/h at 33° north of east

Ship's velocity = 12 mi/h due south

We'll first find the velocity of the airplane with respect to the ground (wind velocity):

Velocity of airplane with respect to ground = Relative velocity of airplane with respect to ship - Ship's velocity

Velocity of airplane with respect to ground = (280 mi/h) at 33° - (12 mi/h) at 180° (south)

Velocity of airplane with respect to ground = (280 mi/h) at 33° - (12 mi/h) at 0°

Velocity of airplane with respect to ground = (280 mi/h) at 33° - 12 mi/h

Velocity of airplane with respect to ground = (280 mi/h) at 33° - (12 mi/h) at 0°

Velocity of airplane with respect to ground = (280 mi/h) at 33° - 12 mi/h = 268 mi/h at 33°

Next, we'll find the velocity of the airplane with respect to the air:

Velocity of airplane with respect to air = Plane's airspeed = 300 mi/h at 30° north of east

Now, we'll find the total velocity of the airplane by adding the velocity of the airplane with respect to the air to the velocity of the airplane with respect to the ground:

Velocity of airplane = Velocity of airplane with respect to air + Velocity of airplane with respect to ground

Velocity of airplane = (300 mi/h) at 30° + (268 mi/h) at 33°

We'll use vector addition to find the resultant velocity:

Velocity of airplane = (300 cos(30°) + 268 cos(33°)) î + (300 sin(30°) + 268 sin(33°)) ĵ

Velocity of airplane ≈ 434.42 î + 295.35 ĵ mi/h

Finding the Wind Speed and Direction:

Wind velocity = Velocity of airplane with respect to air - Velocity of airplane with respect to ground

Wind velocity = (300 mi/h) at 30° - (268 mi/h) at 33°

Wind velocity = (300 cos(30°) - 268 cos(33°)) î + (300 sin(30°) - 268 sin(33°)) ĵ

Wind velocity ≈ 46.42 î - 15.35 ĵ mi/h

The magnitude of the wind velocity is approximately:

|Wind velocity| = √((46.42)^2 + (-15.35)^2) ≈ 49.00 mi/h

The direction of the wind velocity is approximately:

θ = arctan((-15.35) / 46.42) ≈ -18.87° (measured counterclockwise from the positive x-axis)

Thus, the wind is blowing at approximately 49.00 mi/h at an angle of about 18.87° south of east.