Question

A plane can fly 255 miles downwind in the same amount of time as it can travel 205 miles upwind. Find the velocity of the wind (in mph) if the plane can fly 230 mph in still air.

Answer 1

We can think of the wind as a vector of velocity that adds or substract from the velocity of the plane.

When the plane flies downwind, the speed of the wind adds to the speed of the plane.

For a time t the plane can fly 255 miles in this condition. We then can express the average speed as 255/t.

If the speed is the sum of the speed of the plane and the speed of the wind, we can write:

`\bar{v_1}=v_p+v_w=(d_1)/(t)=(255)/(t)_{}`

Now, if the plane flies against the wind, the speed of the wind is substracted from the speed of the plane:

`\bar{v}_2=v_p-v_w=(d_2)/(t)=(205)/(t)`

If we know tha the speed of the plane is vp = 230 mph, we can add both equations and then find t:

`\begin{gathered} v_1+v_2=(255)/(t)+(205)/(t) \\ (v_p+v_w)+(v_p-v_w)=(255+205)/(t) \\ (v_p+v_p)+(v_w-v_w)=(460)/(t)_{} \\ 2v_p=(460)/(t) \\ t=(460)/(2v_p) \\ t=\frac{460\text{ miles}}{2\cdot230\text{ miles per hour}} \\ t=\frac{460\text{ miles}}{460\text{ miles per hour}} \\ t=1\text{ hour} \end{gathered}`

Now that we know the value of t we can use any of the two equations we have written to calculate the speed of the wind (vw):

`\begin{gathered} v_p+v_w=(255)/(t) \\ 230+v_w=(255)/(1) \\ 230+v_w=255 \\ v_w=255-230 \\ v_w=25 \end{gathered}`

Answer: the speed of the wind is 25 miles per hour.