An airplane has an airspeed of 950 km/h due east. There is a crosswind blowing in a direction of 550 south of east at 85 km/h. Calculate the velocity of the airplane relative to the ground.
convert each vector to x-y values
add the x and the y values
convert back to directional form
To calculate the velocity of the airplane relative to the ground, we can use vector addition. The velocity of the airplane relative to the ground will be the vector sum of the airplane's airspeed and the crosswind.
First, we need to break down the velocities into their x and y components:
The airspeed of the airplane, which is 950 km/h due east, has an x-component (horizontal) of 950 km/h and a y-component (vertical) of 0 km/h.
The crosswind blowing south of east at 85 km/h can be broken down into its x and y components using trigonometry. Since the crosswind is 550 south of east, the angle between the crosswind and the x-axis is 90 degrees - 550 = 40 degrees. Using the given magnitude of 85 km/h, we can calculate the x and y components of the crosswind:
x-component = 85 km/h * cos(40 degrees)
y-component = 85 km/h * sin(40 degrees)
Now we can calculate the x and y components of the velocity of the airplane relative to the ground by adding the corresponding components of the airspeed and the crosswind:
x-component of velocity = 950 km/h + x-component of crosswind
y-component of velocity = 0 km/h + y-component of crosswind
Finally, we can find the magnitude and direction of the velocity of the airplane relative to the ground using the Pythagorean theorem and inverse trigonometric functions:
Magnitude = sqrt((x-component of velocity)^2 + (y-component of velocity)^2)
Direction = atan2(y-component of velocity, x-component of velocity)
Calculating these values:
x-component of crosswind = 85 km/h * cos(40 degrees) = 85 km/h * 0.766 = 65.21 km/h
y-component of crosswind = 85 km/h * sin(40 degrees) = 85 km/h * 0.643 = 54.755 km/h
x-component of velocity = 950 km/h + 65.21 km/h = 1015.21 km/h
y-component of velocity = 0 km/h + 54.755 km/h = 54.755 km/h
Magnitude = sqrt((1015.21 km/h)^2 + (54.755 km/h)^2) = sqrt(1030847.0441 + 3005.9425) = sqrt(1033852.9866) = 1016.77 km/h
Direction = atan2(54.755 km/h, 1015.21 km/h) = 3.063 degrees
Therefore, the velocity of the airplane relative to the ground is approximately 1016.77 km/h in a direction 3.063 degrees east of north.