A plane traveling 500 mph (called airspeed) in the direction 120 degrees encounters a wind of 80 mph in the direction of 45 degrees.
What is the resultant direction of the plane (in degrees)?
You do not say if the wind is coming FROM NE or going TOWARD NE
In navigation, sailing, and flying we usually use the FROM convention and your NE breeze has components taking you 80/sqrt 2 south and 80 /sqrt 2 west
However you never know with mathematicians
To calculate the resultant direction of the plane, we can use vector addition. Let's break down the given information into vectors:
1. The airspeed of the plane:
- Magnitude: 500 mph
- Direction: 120 degrees
2. The wind speed:
- Magnitude: 80 mph
- Direction: 45 degrees
To find the resultant vector, we need to add these two vectors together. However, we need to convert the given directions into component form (i.e., x-axis and y-axis).
For the airspeed vector, the x-component can be calculated using the cosine function:
x-component = magnitude * cos(direction)
x-component = 500 mph * cos(120 degrees)
x-component = 500 mph * (-0.5)
x-component = -250 mph
Similarly, the y-component of the airspeed vector can be calculated using the sine function:
y-component = magnitude * sin(direction)
y-component = 500 mph * sin(120 degrees)
y-component = 500 mph * (sqrt(3)/2)
y-component = 250*sqrt(3) mph
For the wind speed vector, we can repeat the same process:
x-component = 80 mph * cos(45 degrees)
x-component ≈ 56.57 mph
y-component = 80 mph * sin(45 degrees)
y-component ≈ 56.57 mph
Now, let's add the x-components and y-components separately to find the resultant vector:
Resultant x-component = -250 mph + 56.57 mph ≈ -193.43 mph
Resultant y-component = 250*sqrt(3) mph + 56.57 mph ≈ 479.43 mph
Using the Pythagorean theorem, we can find the magnitude of the resultant vector:
Resultant magnitude = sqrt((Resultant x-component)^2 + (Resultant y-component)^2)
Resultant magnitude ≈ sqrt((-193.43 mph)^2 + (479.43 mph)^2)
Resultant magnitude ≈ sqrt(37413.44 + 230231.64) ≈ sqrt(267645.08) ≈ 517.38 mph
Finally, to find the resultant direction, we can use the inverse tangent function:
Resultant direction = arctan(Resultant y-component / Resultant x-component)
Resultant direction = arctan(479.43 mph / -193.43 mph) ≈ -68.49 degrees
So, the resultant direction of the plane (in degrees) is approximately -68.49 degrees.