Question

A small plane is flying northeast at a speed of 200 mph. A wind blows due west at 28 mph. Find the resultant speed and direction of the plane.

Answer 1

Answer:

The resultant speed = 181.3 mph

The final direction = 38.7° northeast.

Step-by-step explanation:

We need to find the component in the x-direction and in the y-direction of the speed:

For the plane:

$v_{p_{x}} = 200cos(45) = 141.42$

$v_{p_{y}} = 200sin(45) = 141.42$

For the wind we have:

$v_{w_{x}} = -28$

$v_{w_{y}} = 0$

Now, the total speed in the x-direction and  in the y-direction is:

$V_{x} = v_{p_{x}} + v_{w_{x}} = 141.42 - 28 = 113.42$

$V_{y} = v_{p_{y}} + v_{w_{y}} = 141.42$

Hence, the resultant speed is:

$V = \sqrt{V_{x}^{2} + V_{y}^{2}} = \sqrt{(113.42)^{2} + (141.42)^{2}} = 181.3 mph$

Finally, the direction of the plane is:

$tan(\theta) = \frac{V_{y}}{V_{x}} = \frac{141.42}{113.42} = 1.25$

$\theta = 51.3 ^{\circ}$

$\theta = 90 - 51.3 = 38.7 ^{\circ}$

The plane is moving at 38.7° northeast.

I hope it helps you!