Michael flies in an airplane due north to Michigan. The pilot encounters a direct crosswind from the west of 120 mph. What direction must the pilot steer in order to end up with a resultant of due north and average velocity of 120 mph? Measure and calculate the speed of the airplane
It cant be done.
well, his W directoin component has to be 120, and then a N componetn of 120.
avg velocity=120sqrt2, heading 45 deg W of N.
To determine the direction the pilot must steer and the speed of the airplane, we need to consider vectors.
Let's denote the velocity of the airplane relative to the ground as Vg, which has a magnitude and direction. The velocity of the airplane relative to the air is Va, and the velocity of the wind is Vw. Using these vectors, we can set up an equation.
The resultant velocity (Vr) is the vector sum of the velocity of the airplane relative to the ground (Vg) and the velocity of the wind (Vw):
Vr = Vg + Vw
In this case, we want the resultant velocity (Vr) to be due north. Since the wind is coming from the west, we need to adjust the direction of the airplane to compensate for the crosswind.
If the airplane moves due north with an average velocity of 120 mph, then the magnitude of the resultant velocity (Vr) should also be 120 mph. We can write the equation as:
|Vr| = 120 mph
To calculate the speed of the airplane, we need to find the magnitude of the velocity of the airplane relative to the ground (Vg). Using the Pythagorean theorem:
|Vg|^2 = |Vr|^2 - |Vw|^2
Since we know |Vr| = 120 mph and |Vw| = 120 mph (as given in the question), we can substitute these values into the equation and solve for |Vg|:
|Vg|^2 = (120 mph)^2 - (120 mph)^2
|Vg|^2 = 120^2 mph^2 - 120^2 mph^2
|Vg|^2 = 14400 mph^2 - 14400 mph^2
|Vg|^2 = 0 mph^2
From this calculation, we can see that the magnitude of the velocity of the airplane relative to the ground (|Vg|) is 0 mph. This implies that the airplane is not moving relative to the ground.
Therefore, in order to end up with a resultant of due north and an average velocity of 120 mph, the pilot must steer the airplane directly into the crosswind (westward) with an average speed of 120 mph. However, the airplane will not make any progress towards Michigan since its relative velocity to the ground is zero.