# Determine true velocity?

## For a man walking directly north at 6km/h , it seems that the wind is blowing from the west. To a cyclist that goes 15km/h north the wind seems to be north west. Determine the true velocity of the wind. I am stuck here, From the question I assumed that the wind already has n north(or vertical) component of 6km/h. An I assume that the angle from north west is 45 degrees. I do not know how to determine the angle or the horizontal component . please help with an explanation as to how to find the answer

Jun 21, 2018

I get that the wind comes from the south west quadrant, blowing at a speed of 3 mps (11 km/h) from an angle of 34 degrees south of west.

#### Explanation:

The man walks northwards with the same speed as the wind. So there is a component of the wind velocity that comes from the south in a speed of 6 km/h. He feels a breeze from the west, so there is also a component from the west.

In the case of the cyclist, I think it helps to remember that speed is relative to something. Instead of thinking that the cyclist is moving, we can consider the situation if the cyclist was standing still, and the wind still came from the same direction.

As the cyclist feels the wind from the north west, it means his speed sort of creates the wind. Consider the north-south component of the wind the cyclist feels. That component then would come with a speed of 15-6 km/h = 9 km/h from the north.

As the direction felt is from the north west (${45}^{\circ}$ from pure north and pure west), it follows that the component from west would also have a magnitude of 9 km/h.

Therefore: The two components of the wind would be:
Northwards: 6 kmh
Eastwards: 9 kmh
The actual speed would then be $\sqrt{36 + 81} = \sqrt{117}$ km/h, or 10.816 km/h.

Usually we would give the wind speed in mps, which would mean dividing the above number with 3.600 (as there are 3600 sec. in 1 hour and 1000 m in 1 km). The speed, therefore, would be $\frac{10.816}{3.600} = 3.004$ mps or rounded to 3 mps.

The angle of the speed from the x axis would be arctan (6/9)=33.69 degrees, or rounded to 34 degrees, so from a direction of 34 degrees south of west.