An airplane's velocity with respect to the air is 580 miles per hour, and its bearing is 332 degrees. The wind, at the altitude of the plane, is from the SW and has a velocity of 60 mph. What is the true direction of the plane, and what is the speed with respect to the ground?
580 at 332° = (-272,512)
60 from SW = 60 at 225° = (-42,-42)
add them up to get (-314,470) = 565 at 326°
p'jn
To determine the true direction of the plane and its speed with respect to the ground, we need to consider the wind vector and use vector addition.
Let's break down the given information:
1. Airplane velocity with respect to the air: 580 mph (given)
2. Airplane bearing: 332 degrees (given)
3. Wind velocity: 60 mph (given)
4. Wind direction: From the SW (Southwest)
To find the true direction of the plane, we need to find the resultant of the airplane velocity vector and the wind velocity vector.
1. Convert the bearing to a heading angle: Since the bearing is given as 332 degrees, we subtract it from 360 to find the heading angle.
Heading angle = 360 - 332 = 28 degrees
2. Convert the heading angle to a wind-relative angle: Since the wind is from the SW, we subtract the heading angle from 180.
Wind-relative angle = 180 - 28 = 152 degrees
3. Use trigonometry to find the components of the airplane velocity vector:
Velocity in the eastward direction (VE) = Velocity × cos(Wind-relative angle) = 580 × cos(152)
Velocity in the northward direction (VN) = Velocity × sin(Wind-relative angle) = 580 × sin(152)
4. Calculate the components of the wind velocity vector:
Wind velocity in the eastward direction (WE) = Wind velocity × cos(225 degrees) = 60 × cos(225)
Wind velocity in the northward direction (WN) = Wind velocity × sin(225 degrees) = 60 × sin(225)
5. Find the resultant vector by adding the respective components of the airplane velocity and the wind velocity vectors:
True velocity in the eastward direction = VE + WE
True velocity in the northward direction = VN + WN
6. Calculate the magnitude of the true velocity vector using the Pythagorean theorem:
True speed = sqrt((True velocity in the eastward direction)^2 + (True velocity in the northward direction)^2)
7. Determine the true direction of the plane using trigonometry:
True direction = arctan(True velocity in the eastward direction / True velocity in the northward direction)
Calculating the above steps will give us the true direction of the plane and its speed with respect to the ground.
To find the true direction of the plane and its speed with respect to the ground, we need to use vector addition.
Let's break down the velocities into their components:
1. The velocity of the plane with respect to the air (air velocity):
- Velocity magnitude = 580 miles per hour
- Velocity bearing = 332 degrees
2. The velocity of the wind:
- Velocity magnitude = 60 mph
- Velocity direction = Southwest (SW)
To find the true direction of the plane, we need to add the vectors of the air velocity and wind velocity.
Step-by-step procedure:
1. Convert the bearing of the air velocity to a unit vector:
- Start with a vector pointing to the right (x-axis) with a length of 1.
- Rotate the vector counterclockwise by 332 degrees.
- This will give us a unit vector that represents the direction of the air velocity.
2. Multiply the air velocity unit vector by the air velocity magnitude (580 mph) to get the air velocity vector.
3. Convert the wind velocity from mph to a velocity vector by multiplying the wind velocity magnitude by the unit vector in the direction of the wind (southwest).
4. Add the air velocity vector and wind velocity vector together to get the resulting velocity vector of the plane with respect to the ground.
5. Determine the direction of the resulting velocity vector using trigonometry.
6. Determine the magnitude (speed) of the resulting velocity vector using the Pythagorean theorem.
Let's calculate it step by step:
Step 1: Convert the bearing of the air velocity to a unit vector.
To convert the bearing to a unit vector, we can use trigonometry. We'll use the sine and cosine functions to determine the x and y components of the vector.
- Bearing angle = 332 degrees
To get the x-component (horizontal component):
x = cos(angle) = cos(332 degrees)
To get the y-component (vertical component):
y = sin(angle) = sin(332 degrees)
Step 2: Multiply the air velocity unit vector by the air velocity magnitude.
Air velocity vector = (x-component, y-component) * air velocity magnitude
Step 3: Convert the wind velocity from mph to a velocity vector.
Since the wind is blowing from the southwest, the unit vector representing the wind direction will have components (-1, -1). Multiply this unit vector by the wind velocity magnitude to get the wind velocity vector.
Wind velocity vector = (-1, -1) * wind velocity magnitude
Step 4: Add the air velocity vector and wind velocity vector.
Resulting velocity vector = Air velocity vector + Wind velocity vector
Step 5: Determine the direction of the resulting velocity vector.
To find the direction (angle) of the resulting velocity vector, we can use the arctan function. The arctan of the y-component divided by the x-component will give us the angle in radians.
Step 6: Determine the magnitude (speed) of the resulting velocity vector.
Using the Pythagorean theorem, the magnitude of the resulting velocity vector (speed with respect to the ground) is given by:
Magnitude = sqrt(x^2 + y^2)
By following these steps, we can calculate the true direction of the plane and its speed with respect to the ground.