# Heuristic derivation of air resistance - why should the velocity of the air be equal to the velocity of the car?

David MacKay gives a heuristic derivation of air resistance of a driving car in his book Sustainable energy without the hot air.

He starts with a tube of air with the cross sectional area of the car $$A_{\text{car}}$$ and the length $$d$$ which is the distance, the car is moving with constant velocity $$v$$.

He estimates the power which the car transfers to the air tube as the kinetic energy of the air tube due to the interaction with the car divided by the time it takes the car to cross it. For the kinetic energy, he makes the assumption that the velocity of the air tube is equal to the velocity $$v$$ of the car.

My question is: How may this assumption be justified? I tried to make a simple model of the massive car colliding with a single lightweight air molecule but in this case, the standard formula for elastic collisions implies that the air molecule has a final velocity of $$2v$$.

(I'm aware that I'm ignoring the drag coefficiant so far. If we take the car to be a cube, it should be very nearly equal to $$1$$. Also, MacKay pictures its influence as effectively shrinking the cross-sectional area of the air tube and not as influencing its velocity. I'd prefer to have an answer from within this framing but I'm also open to arguments why other framings are better.)

• If the analysis is done in the rest frame of the car, then the air would have a velocity $v$. Is this what is being done? – Aaron Stevens Sep 19 at 13:31
• That's a good point. I don't think it is what he does (the link in my question directly points to his calculation). But actually I'm confused how to analyze this in different reference frames. If I consider the car to be at rest, the air tube has an initial velocity of $-v$ and a final velocity of $+v$. So only its momentum changes but not its kinetic energy. If no kinetic energy is transferred to the air and the velocity of the car is 0 as well, how do we even get a non-zero power? (surely the fuel consumption of the car can't depend on the reference frame) – Marc Sep 19 at 13:47
• In Newtonian mechanics the measure of KE of a system does depend on the reference frame. Really, the link is just a way to get a rough estimate of the power. If you try to get too much detail from it, the basic assumptions don't make sense physically. – alephzero Sep 19 at 14:37
• Regarding the frame dependence: The inner energy of the fuel doesn't depend on the reference frame. Energy is conserved, so the rate of energy transfer, i. e. the power, shouldn't depend on the reference frame either. My problem in the comment above ist that I can't see how the power is non-zero in the restframe of the car. – Marc Sep 19 at 14:56