My Snow and Wind Problem

You are driving your car at speed 100 km/hr, and the snow seems to be blowing from the right front of your car, at an angle of 30° (30 degrees). You stop your car, and you find that the snow is really blowing from an angle of 60° (60 degrees). See the diagram, on the left. What is the speed of the wind?

Whether you attempt to solve it or not, you may enjoy the following approach to the problem.

This is a much simpler version of the above problem. You are driving your car 100 km./hr. This time the snow seems to be blowing from the right front of your car, at an angle of 45° (45 degrees). You stop your car, and you find that the snow is really blowing directly from the right side, 90° (90 degrees) from the direction that your car is pointed. See the second diagram, on the left. What is the velocity (speed and direction) of the wind?

This is now a simple problem, which invites the use of vectors. If you have never used vectors before, don't worry; this is pretty easy. We just draw arrows to represent the speed and direction of the wind. See the diagram on the right. Instead of considering the car to be moving forward at 100 km./hr., I will pretend that the car is standing still and the wind is moving diagonally at 45° (45 degrees). We don't know how fast the wind is moving at that angle. But, we do know part of its velocity, the "component" opposite to the direction your car is pointed. That is 100 km./hr. The other component (x) is also 100 km./hr. When you stop your car, x is exactly the speed of the wind.

So, we have solved this simpler problem. The velocity of the wind is 100 km./hr., directly from the right of your car.

You may want to retry the original problem, before going on the next part of the explanation.

Back to our original problem, let's use letters instead of numbers. Once everything is drawn, as vectors (see the diagram at the left), we just have a trigonometry problem. Here, w is the speed that we are trying to find. A is its angle from the direction that your car is pointing. The trig:

```
w = x cos B
h = x sin B
h = (x+z) sin A
x sin B = x sin A + z sin A
x (sin B - sin A) = z sin A
x = (z sin A)/(sin B - sin A)
w = (z sin A cos B)/(sin B - sin A)```

Plugging in z=100 km/hr, cos 60°=0.5, sin 60°=0.8660, and sin 30°=0.5, we get w=136.6 km/hr.