An airplane is flying in a direction of 65 degrees at 450 mph. A wind is blowing in a direction of 20 degrees at 35 mph. Find the actual speed and direction of the plane.
You don't say whether these are courses from due North, or from the positive x-axis. I'll do the latter.
the x- and y-components of the vectors are
(450 cos65°, 450 sin65°) = (190.18,407.84)
(35 cos20°, 35 sin20°) = (32.89, 11.97)
Add them up to get
(223.07,410.81) = 475 mph at 61°
To find the actual speed and direction of the plane, we can use vector addition.
Step 1: Separate the velocities into horizontal and vertical components.
The velocity of the plane can be separated into horizontal and vertical components as follows:
Horizontal component of the plane's velocity = speed of plane * cosine(direction of plane)
Vertical component of the plane's velocity = speed of plane * sine(direction of plane)
Horizontal component of the plane's velocity = 450 mph * cos(65°)
Vertical component of the plane's velocity = 450 mph * sin(65°)
Step 2: Add the wind velocity vector to the plane's velocity vector.
The horizontal component of the wind velocity can be calculated as:
Horizontal component of the wind velocity = speed of wind * cosine(direction of wind)
Horizontal component of the wind velocity = 35 mph * cos(20°)
The vertical component of the wind velocity can be calculated as:
Vertical component of the wind velocity = speed of wind * sine(direction of wind)
Vertical component of the wind velocity = 35 mph * sin(20°)
Step 3: Add the horizontal and vertical components of the plane's velocity to the horizontal and vertical components of the wind velocity.
The horizontal component of the actual velocity = Horizontal component of the plane's velocity + Horizontal component of the wind velocity
The vertical component of the actual velocity = Vertical component of the plane's velocity + Vertical component of the wind velocity
Step 4: Calculate the magnitude and direction of the actual velocity.
The magnitude of the actual velocity can be found using the Pythagorean theorem:
Magnitude of the actual velocity = √(actual velocity-horizontal^2 + actual velocity-vertical^2)
The direction of the actual velocity can be found using the inverse tangent function:
Direction of the actual velocity = arctan(actual velocity-vertical / actual velocity-horizontal)
By following these steps, you can find the actual speed and direction of the plane.