Question

Flying against the wind, an airplane travels 5100 kilometers in 6 hours. Flying with the wind, the same plane travels 3870 kilometers in 3 hours. What is the rate of the plane in still air and what is the rate of the wind?

Answer 1

The rate of the plane in still air is 1070 km/h and what is the rate of the wind is 220 Km/h

Explanation:

Relative Speeds

If an object moves at a speed
`v_p` with respect to another object at a speed
`v_w`, the relative speeds between them can be
`v_p+v_w` or
`v_p-v_w` depending if they are collaborative or one against the other.

The speed of an object who travels a distance x in a time t is

`\displaystyle v=(x)/(t)`

We know that an airplane travels 5100 kilometers in 6 hours when flying against the wind, and travels 3870 kilometers in 3 hours when flying with the wind. Let's call
`v_p` and
`v_w` the speeds of the plane in still air and the wind, respectively. The first travel is performed by the plane with a wind whose speed subtracts from its own, so the relative speed is

`v_p-v_w=\displaystyle (5100)/(6)`

`v_p-v_w=850`

The second travel is performed with the wind pushing in the same direction, so

`\displaystyle v_p+v_w=(3870)/(3)`

`v_p+v_w=1290`

Adding both equations, we have

`v_p-v_w+v_p+v_w=850+1290`

Simplifying and solving

`2v_p=850+1290`

`v_p=1070`

Replacing into the second equation

`1070+v_w=1290`

`v_w=1290-1070=220`

The rate of the plane in still air is 1070 km/h and what is the rate of the wind is 220 Km/h