The velocity of an airplane with respect to the ground is 200 m/s at an angle of 30 degrees NORTH of EAST. The velocity of the airplane with respect to the air is 150 m/s at an angle of 60 degrees NORTH of EAST. What is the velocity of the air with respect to the ground?
Wind speed vector + Air speed vector = Ground speed vector
Perform the vector subtraction.
Wind speed = Ground speed - Air speed
North component of wind speed
= 200 sin30 -150 sin60 = ?
East component of wind speed
= 200 cos30 -150 cos60 = ?
yep and then you do the sqrt( -29.9^2 +98.2^2) to get the velocity
To determine the velocity of the air with respect to the ground, we can subtract the velocity of the airplane with respect to the ground from the velocity of the airplane with respect to the air.
Given:
Velocity of the airplane with respect to the ground (Vg) = 200 m/s at an angle of 30 degrees NORTH of EAST
Velocity of the airplane with respect to the air (Va) = 150 m/s at an angle of 60 degrees NORTH of EAST
Step 1: Resolve the velocities into their horizontal and vertical components.
The horizontal component of the velocity can be determined using the cosine function, while the vertical component can be determined using the sine function.
For the velocity of the airplane with respect to the ground:
Vg(horizontal) = Vg * cos(30°)
Vg(vertical) = Vg * sin(30°)
For the velocity of the airplane with respect to the air:
Va(horizontal) = Va * cos(60°)
Va(vertical) = Va * sin(60°)
Step 2: Subtract the horizontal and vertical components of the airplane's velocity with respect to the ground from the horizontal and vertical components of the airplane's velocity with respect to the air.
Velocity of the air with respect to the ground:
Vair(horizontal) = Va(horizontal) - Vg(horizontal)
Vair(vertical) = Va(vertical) - Vg(vertical)
Step 3: Calculate the magnitude and angle of the air velocity with respect to the ground.
The magnitude can be determined using the Pythagorean theorem:
Vair(magnitude) = sqrt((Vair(horizontal))^2 + (Vair(vertical))^2)
The angle can be determined using the inverse tangent function:
Angle = atan(Vair(vertical) / Vair(horizontal))
By substituting the calculated values into the equations, we can determine the velocity of the air with respect to the ground.
To find the velocity of the air with respect to the ground, we need to subtract the velocity of the airplane with respect to the ground from the velocity of the airplane with respect to the air.
Step 1: Convert the given velocities into their horizontal and vertical components using trigonometry.
For the velocity of the airplane with respect to the ground, we have:
Velocity_ground_x = 200 m/s × cos(30°) = 200 m/s × √3/2 ≈ 346.4 m/s (towards the east)
Velocity_ground_y = 200 m/s × sin(30°) = 200 m/s × 1/2 = 100 m/s (towards the north)
For the velocity of the airplane with respect to the air, we have:
Velocity_air_x = 150 m/s × cos(60°) = 150 m/s × 1/2 = 75 m/s (towards the east)
Velocity_air_y = 150 m/s × sin(60°) = 150 m/s × √3/2 ≈ 129.9 m/s (towards the north)
Step 2: Subtract the velocity components of the airplane relative to the ground from the velocity components of the airplane relative to the air.
Velocity_air_relative_to_ground_x = Velocity_air_x - Velocity_ground_x = 75 m/s - 346.4 m/s ≈ -271.4 m/s (towards the west)
Velocity_air_relative_to_ground_y = Velocity_air_y - Velocity_ground_y = 129.9 m/s - 100 m/s ≈ 29.9 m/s (towards the north)
Therefore, the velocity of the air with respect to the ground is approximately 271.4 m/s towards the west and 29.9 m/s towards the north.