what is the average velocity of the plane in still​ air?

Question

A plane flies 400 miles with the wind and 320 miles against the wind in the same length of time. If the average velocity of the wind is 27 ​mph

Answer 1

Let,

s₁ = 400 miles be the distance travelled with the wind,

s₂ = 320 miles be the distance travelled against the wind,

v_w = 27 mph be the velocity of the wind with respect to the ground,

v_p (to be found) be the velocity of the plane with respect to the air,

or equivalently the velocity of the plane with respect to the ground in still air.

With the wind, the velocity of the plane with respect to the ground is

v_p+v_w

Against the wind, the velocity of the plane with respect to the ground is

v_p-v_w

In general, time traveled is the ration between distance travelled and average velocity, i.e.,

s/v

The statement that the plane took the same time to travel both distances is expressed by

s₁/(v_p+v_w) = s₂/(v_p-v_w)

From that we can solve

v_p = v_w(s₁+s₂)/(s₁-s₂)

Substituting actual numbers

v_p = 27 x 720/80 = 243 mph