An airplane has an airspeed of 450 kilometers per hour bearing Upper N45E. The wind velocity is 60 kilometers per hour in the direction Upper N30W. Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? What is its direction?
Upper?
plane
... N component ... 450 kph cos(45º)
... E component ... 450 kph sin(45º)
wind
... N component ... 60 kph cos(30º)
... W component ... 60 kph sin(30º)
add the components to find the resultant vector
To find the resultant vector representing the path of the plane relative to the ground, we need to add the vectors of the airplane's airspeed and the wind velocity.
First, let's break down the given information into its components. The airspeed of the plane has a magnitude of 450 kilometers per hour and a bearing of N45E, which means it is pointing 45 degrees east of north. The wind velocity has a magnitude of 60 kilometers per hour and a bearing of N30W, which means it is pointing 30 degrees west of north.
Next, we need to convert the given information into vector form. To do this, we can use trigonometry. We can represent the airspeed vector as A, with magnitude 450 km/h and direction 45 degrees east of north. We can represent the wind velocity vector as W, with magnitude 60 km/h and direction 30 degrees west of north.
Now, we can find the x and y components of the vectors A and W:
Ax = 450 * cos(45°) (east is positive)
Ay = 450 * sin(45°) (north is positive)
Wx = 60 * sin(30°) (west is negative)
Wy = 60 * cos(30°) (north is positive)
Now, we can add the components of A and W:
Rx = Ax + Wx
Ry = Ay + Wy
To find the magnitude of the resultant vector, we can use the Pythagorean theorem:
R = sqrt(Rx^2 + Ry^2)
Finally, we can find the direction of the resultant vector using trigonometry:
θ = atan(Ry / Rx)
Calculating the values:
Ax = 450 * cos(45°) = 318.19 km/h
Ay = 450 * sin(45°) = 318.19 km/h
Wx = 60 * sin(30°) = 30 km/h
Wy = 60 * cos(30°) = 51.96 km/h
Rx = Ax + Wx = 318.19 + 30 = 348.19 km/h
Ry = Ay + Wy = 318.19 + 51.96 = 370.15 km/h
R = sqrt(Rx^2 + Ry^2) = sqrt(348.19^2 + 370.15^2) = 494.23 km/h
θ = atan(Ry / Rx) = atan(370.15 / 348.19) = 48.11 degrees
Therefore, the ground speed of the plane is 494.23 km/h and its direction is N48E.