# drag force relationship with cube of speed?

So I saw this physics website that said drag grows quadratically with speed. Meaning the drag force at 200 km/h is 4 times what it is at 100 km/h. Ok I'm fine with that. But then it said the work required to accelerate to that speed grows cubically. Meaning I need 8 times the power to accelerate from 100 km/h to 200km/h? this part I dont get. So if my imaginary engine was producing x kilowatts or kilonewtons, and I pressed a button and its now doing 4x:

• Isnt it logical that I should accelerate if I am going only 100 km/h, since drag isnt 4x yet??
• Shouldnt I keep accelerating until equilibrium out at 200 km/h?

Update: the power vs force distinction solves it. I was mislead by the use of the word "accelerate" in that page. I thought it was discussion about drag and acceleration. There is no accelerate. The power required to go from A to B in half the time is 2x larger. And drag grows quadratically.

You seem to be confusing force (thrust or drag) with energy (under which concept work and power comes). Don't. You'll be spared much grief.

Work done is force times distance.

$$W = Fs$$

Power is work done divided by time

$$P = W/t$$

When you double velocity, force quadruples. As does work.

$$W_\text{new} = 4Fs = 4W$$

Time halves as well, with the following effect on power.

$$P_\text{new} = 4W/0.5t = 8P$$

• Whoever just edited. Make a superficial edit again so your name can be seen. You're the victim of an unfortunate coincidence Commented Feb 11, 2021 at 8:31
• No need to see my name (it is still in the edit history) ;) Commented Feb 11, 2021 at 8:35
• power vs force solves it. I was missing the distinction. yes drag force goes up with the square, but power goes up another 2x because times goes down 2x.
– kaz
Commented Feb 11, 2021 at 9:00