Question

A plane flies on a bearing of 120 degrees at a constant speed of 550 km/h. If the velocity of the wind is 50 km/h on a bearing 220 degrees, what is the velocity of the plane with respect to the ground?

Textbook Answer: 543.5 km/h at a bearing of 125.2 degrees

My answer: 543.6 km/h at a bearing of 175 degrees

My work:

Let vector w represent 50 km/h
Let vector v represent 550 km/h
Let vector r represent the resultant

I'm using component vectors. . .

w = (50cos220 , 50sin220)
w = (-38.3 , -32.1)

v = (550cos120 , 550sin120)
v = (-275 , 476.3)

r = (-38.3 , -32.1) + (-275 , 476.3)
r = (-313.3 , 444.2)

To find the resultant:
| r | = (-313.3)^2 + (444.2)^2
| r | = (sqrt)295 470.53
| r | = 543.6 km/h

To find the angle:
tanx = 444.2/-313.3
x = 55 degrees

120 + 55 = 175 (To find the bearing)

----------------What did I do wrong? How can I get the 125 degree instead of 175 degrees?

Answer 1

The reason why you are getting 175 degrees is because you are solving for the wrong angle. You are looking for the angle between the resultant and x - axis (-313.3) that is relative to the ground.

cosx = adj. / hypt.
x = cos- (-323.3/543.6)
x = 125 degrees

Answer 2

if you take the archtan of 444.2/-313= -54.80
than +180 then it will be 125.19