Question

A plane has an airspeed of 400 mph . The pilot wishes to reach a destination 900 mi due east, but a wind is blowing at 60 mph in the direction 40 ∘ north of east.

In what direction must the pilot head the plane in order to reach her destination?

How long will the trip take?

Answer 1

The pilot needs to head in a direction south of east to offset the north east wind that is blowing.

Use vectors.

For the wind the bearing is 40° north of east.
w = 60(icos40° + jsin40°) = 60(0.77i + 0.64j)

For the plane the bearing is unknown south of east. Use θ.
p = 400(icosθ + jsinθ)

The resultant path of the plane in the wind is p + w and is due east. This means the j component nets out to zero.

p + w =60(0.77i + 0.64j) + 400(icosθ + jsinθ)
p + w = (46.2 + 400cosθ)i + (38.4 + 400sinθ)j


38.4 + 400sinθ = 0
400sinθ = -38.4
sinθ = -0.096
θ = arcsin(-0.096) ≈ -5.50887°

The pilot must head 5.50887° south of east.
___________________

cosθ = √(1 - sin²θ) = √(1 - 1474.56/160000) = √(0.990784)= 0.9954

Plug in for sinθ and cosθ.

p + w = (46.2 + 400x0.9954)i + (38.4 + 400x(-0.096))j
p + w = (46.2 + 398.16)i + (38.4 -38.4)j
p + w = 444.36i + 0j
p + w = 444.36i

The length of time to travel 900 miles due east is:

t = 900 /444.36 ≈ 2.025384823 hours
t ≈ 2 hours, 1 minutes, 31.3853 seconds