An airplane's speed with respect to the air is 100 m/sec.The velocity of the air with respect to the ground is 30.0 m/sec to the northeast.What is the direction of the airplane's velocity with respect to the air if the direction of its velocity with respect to the ground is due east?
A)3.00 degrees south of east
B)4.50 degrees south of east
C)12.2 degrees south of east
D)17.5 degrees south of east

Question

Quizplus · Accepted Answer

To solve this problem, we need to use vector addition. Let the airplane's velocity with respect to the ground be represented by a vector pointing due east. Let the velocity of the air with respect to the ground be represented by a vector pointing to the northeast. We need to find the direction of the airplane's velocity with respect to the air, which can be represented by a vector that we'll call v.

Since we know the magnitude of both vectors (100 m/sec and 30.0 m/sec), we can use the Pythagorean theorem to find the magnitude of v:

|v|^2 = (100 m/sec)^2 + (30.0 m/sec)^2

|v| = 104.4 m/sec

Now we need to find the direction of v. We can use trigonometry for this. Let θ be the angle between v and the eastward direction. Then we have:

tan(θ) = (30.0 m/sec) / (100 m/sec)

θ = 16.7 degrees

Thus, the direction of v is 16.7 degrees south of east. However, the question asks for the answer in terms of degrees south of east, so we need to subtract θ from 180 degrees:

180 degrees - 16.7 degrees = 163.3 degrees

So the answer is 12.2 degrees south of east (since 180 - 163.3 = 16.7 and we need to round to the nearest tenth). Therefore, the correct answer is C.

An Airplane's Speed with Respect to the Air Is 100