Question

At low speeds, Drag force is proportional to speed due to Stokes law.

At higher speeds, Drag is proportional to the square of speed. While modelling the speed of a water rocket, you realise that when there was inherently high drag in the system - say the coefficient of drag is 1.00 - the model fit very well when you placed drag force proportional to v². However, when there was inherently lower drag in the system - say the coefficient of drag is 0.5 - the model doesn't fit well for v2v2. Instead, interestingly enough, it fits very well for values such as v2.8v2.8orv3v3. Now, is it reasonable to argue that the reason this happens is because of the following:
A. When there is inherently higher drag, the rocket attains lower speeds, and hence is proportional to lower powers of velocity.
B. Similarly, when there is inherently lower drag, the system attains higher speeds and therefore is proportional to higher powers of velocity.
So, is this argument reasonable? If yes how would you make the units consistent?

Answer 1

At low speeds, drag force is proportional to speed according to Stokes' law. At higher speeds, drag force is proportional to the square of speed. The argument that higher drag leads to lower speeds and lower powers of velocity, and vice versa, is reasonable.

Step-by-step explanation:

Drag force is influenced by various factors, including the speed of an object and the coefficient of drag. At low speeds, the drag force is proportional to the speed, following Stokes' law. At higher speeds, the drag force is proportional to the square of the speed.

Therefore, it is reasonable to argue that when there is inherently higher drag in the system, such as a coefficient of drag of 1.00, the model fits well when the drag force is proportional to v². On the other hand, when there is inherently lower drag, such as a coefficient of drag of 0.5, the model fits better for values such as v².8 or v³.

This is because when there is higher drag, the object attains lower speeds, thus the drag force is proportional to a lower power of velocity. Conversely, when there is lower drag, the object attains higher speeds, resulting in the drag force being proportional to a higher power of velocity.

In order to make the units consistent, you can assign appropriate values to the coefficients of drag and velocity. For example, if you use v², you might need to adjust the coefficient of drag to maintain the correct units.