A plane if flying at 200 mph with a heading of 45 degrees and encounters a wind of 100 mph from the west. What is the resulting velocity and heading?
I just did this for Sarah below :(
Trig - Damon, Sunday, March 3, 2013 at 6:24pm
East speed = 100 + 200 sin 45 = 241
North speed = 200 cos 45 = 141
speed = sqrt( 141^2 + 241^2) = 279 mph
tan east of north = 241/141
so angle east of north = 60 degrees
To find the resulting velocity and heading of the plane after encountering the wind, we need to use vector addition.
Step 1: Convert velocities to vectors
The plane's velocity can be represented as a vector with a magnitude of 200 mph and a direction of 45 degrees. The wind's velocity can also be represented as a vector with a magnitude of 100 mph and a direction of 270 degrees (from the west).
Step 2: Resolve vectors into components
To perform vector addition, we need to break down the velocities into their horizontal (east-west) and vertical (north-south) components. We can use trigonometry to do this.
The plane's velocity components can be calculated as:
Vx = velocity * cos(angle)
Vy = velocity * sin(angle)
Substituting in the values:
Vx = 200 mph * cos(45 degrees)
Vy = 200 mph * sin(45 degrees)
Vx ≈ 141.4 mph
Vy ≈ 141.4 mph
Similarly, the wind's velocity components are:
Vx = 100 mph * cos(270 degrees)
Vy = 100 mph * sin(270 degrees)
Vx = 0 mph
Vy = -100 mph
Step 3: Perform vector addition
Now that we have the components of the velocities, we can add them together to find the resulting velocity.
Resultant Vx = Plane's Vx + Wind's Vx
= 141.4 mph + 0 mph
= 141.4 mph
Resultant Vy = Plane's Vy + Wind's Vy
= 141.4 mph + (-100 mph)
= 41.4 mph
Step 4: Find the magnitude and direction of the resultant velocity
To find the magnitude of the resultant velocity, we can use the Pythagorean theorem:
Resultant velocity = √((Resultant Vx)^2 + (Resultant Vy)^2)
= √((141.4 mph)^2 + (41.4 mph)^2)
≈ 149.1 mph
To find the direction, we can use trigonometry:
Resultant angle = arctan(Resultant Vy / Resultant Vx)
= arctan(41.4 mph / 141.4 mph)
≈ 16.9 degrees
Therefore, the resulting velocity of the plane is approximately 149.1 mph with a heading of approximately 16.9 degrees.