Difficult Velocity/Travel/Wind problem

[Aaron7]

Homework Statement

An aeroplane makes a round trip between two airports A and B located on the same line of longitude. On a windless day the aeroplane travels with constant speed v and the round trip time is T0. On the following day the same journey is made but there is a wind blowing at speed kv (k<1) at an angle z to the line AB. The pilot maintains the same speed v relative to the air and always flies along the line AB and BA. The round trip time is now T (neglecting take offs, landings and stopovers)

Find T/T0

Homework Equations

See below

The Attempt at a Solution

I drew a vector diagram for velocity with v_r going horizontally(new speed along AB when there is wind) and the components v (plane speed) and kv (wind speed).

I then used the cosine rule for v^2 = v_r^2 + (kv)^2 - (2 v_r kv cosθ)
Solved for v_r to get:
v_r = kvcosθ + v√(1-k²sinθ)

Now, T/T0 = v/v_r ,
so T/T0 = 1/(kvcosθ + v√(1-k² sinθ))

-----------------------------------------------------------------------
The answer is given to be T/T0 = [(1-k²(sin²θ))^0.5]/(1-k²)
-----------------------------------------------------------------------

I have tried to rearrange to get the same answer but I think I have gone wrong somewhere.

Thanks for the help.

 

[tiny-tim]

Hi Aaron7!

An aeroplane makes a round trip between two airports A and B located on the same line of longitude.

(what does "r" stand for? )

There are two trips, AB and BA, with different speeds.

 

[Aaron7]

I got it Thanks for the help.