A plane is flying at 240 mph heading North 60 degrees East. The wind is blowing at South 30 degrees East at 30 mph.
What is the largest and smallest angles?
Vp = 240mi/h, N60E = 30deg CCW.
Vw = 30mi/h, S30E = 300deg. CCW.
Add all hor components of velocity; and
then add all ver. components.
X = hor = 240cos30 + 30cos300 =
207.8 + 15 = 222.8mi/h.
Y = ver = 240sin30 + 30sin300 =
120 - 26 = 94mi/h.
Now we have a single rt triangle with
X and Y representing the hor and ver
side, respectively.
tanA = Y/X = 94/222.8 = 0.4219,
A = 22.9deg.,CCW = N67.1E. = Bearing of
the plane.
The largest angle = 90deg.
The smallest angle = 22.9deg.,CCW.
To find the largest and smallest angles in this situation, we need to consider the velocity vectors of the plane and the wind.
Let's start by breaking down the velocity of the plane into its horizontal (East-West) and vertical (North-South) components. Since the plane is flying at 240 mph in a direction of North 60 degrees East, we can use trigonometry to find these components:
Horizontal component of the plane's velocity:
240 mph * cos(60 degrees) = 120 mph
Vertical component of the plane's velocity:
240 mph * sin(60 degrees) = 207.9 mph (rounded to one decimal place)
Now, let's do the same thing with the wind velocity. The wind is blowing at 30 mph in a direction of South 30 degrees East:
Horizontal component of the wind's velocity:
30 mph * cos(30 degrees) = 25.98 mph (rounded to two decimal places)
Vertical component of the wind's velocity:
30 mph * sin(30 degrees) = 15 mph
Next, we need to calculate the resulting horizontal and vertical velocities of the plane when accounting for the wind.
Resultant horizontal velocity of the plane:
Horizontal velocity of the plane + Horizontal velocity of the wind:
120 mph + 25.98 mph = 145.98 mph (rounded to two decimal places)
Resultant vertical velocity of the plane:
Vertical velocity of the plane - Vertical velocity of the wind:
207.9 mph - 15 mph = 192.9 mph (rounded to one decimal place)
Now that we have the resultant horizontal and vertical velocities of the plane, we can find the angle that the plane is flying relative to the North direction using trigonometry.
Angle = arctan(Vertical velocity / Horizontal velocity)
Angle = arctan(192.9 mph / 145.98 mph)
Using a calculator or a trigonometric table, we find that the angle is approximately 51.74 degrees (rounded to two decimal places).
To find the largest and smallest angles, we consider the angle between the plane and the North direction. The largest angle occurs when the plane is heading directly North, which is 0 degrees. The smallest angle occurs when the plane is heading directly South, which is 180 degrees.
Thus, the largest angle is 0 degrees, and the smallest angle is 180 degrees.