How to Calculate Velocity Relative to Air: Explained with Examples

[nesan]

Homework Statement

Sorry, I just have a general question. I know how to calculate velocity of anything relative to the ground, but how do you calculate the velocity of something relative to the air (?) if the velocity of the wind is given.

Please explain, with a small example if possible. Thank you. ^_^

2. What I have so far

For example let's say a plane was flying at a velocity of 10 m/s north and the wind was blowing 5 m/s east. I know the velocity of the plane relative to the ground would be sqrt(10^2 + 5^2)[N x Degree E], but how does this work if you have to find the velocity relative to the air?

 

[cepheid]

I'll use subscripts, p, a, and g for plane, air, and ground respectively. A comma between two of them. such as a,g, means "air relative to ground." So, as you know, it goes:

$$\vec{v}_{p,g} = \vec{v}_{p,a} + \vec{v}_{a,g}$$

The velocity of the plane relative to the ground is equal to the velocity of the plane relative to the air, PLUS the velocity of the air relative to the ground.

So, in this case, $v_{p,a}$ is clearly just 10 m/s N. Think about it. If you were in a river that ran east-west, and the current was 5 m/s east, and you just sat there and allowed yourself to be carried along with the current, then your velocity relative to the river would be 0 (because you are moving along with it). Your velocity relative to the bank would be 5 m/s east.

If you then began to swim across the river at a velocity of 10 m/s north, then your velocity relative to the water, instead of being 0, would now be 10 m/s north.

It's the same idea with the plane example.

 

[nesan]

I'll use subscripts, p, a, and g for plane, air, and ground respectively. A comma between two of them. such as a,g, means "air relative to ground." So, as you know, it goes:

$$\vec{v}_{p,g} = \vec{v}_{p,a} + \vec{v}_{a,g}$$

The velocity of the plane relative to the ground is equal to the velocity of the plane relative to the air, PLUS the velocity of the air relative to the ground.

So, in this case, $v_{p,a}$ is clearly just 10 m/s N. Think about it. If you were in a river that ran east-west, and the current was 5 m/s east, and you just sat there and allowed yourself to be carried along with the current, then your velocity relative to the river would be 0 (because you are moving along with it). Your velocity relative to the bank would be 5 m/s east.

If you then began to swim across the river at a velocity of 10 m/s north, then your velocity relative to the water, instead of being 0, would now be 10 m/s north.

It's the same idea with the plane example.

Thank you very much, just what I was looking for. Very helpful. ^_^ Thank you.