We did an experiment where we changed only the mass of the coffee filters and then measured the terminal velocity. We got that the relationship is proportional. Why is that?

You can check the answer of the people under the question at Quora “drag coefficient of coffee filter“

The data exhibits a zone of linear resistance.

To better discuss the findings, it can be illustrated by (one at the time):

Changing the filter’s pore size; smaller.

Continue increasing the filter’s thickness.

Increasing the coffee concentration.

The air resistance remains effectively constant with the additional coffee filters stacked together, because the surface area in contact with air remains constant.

The gravitational force on the stack of coffee filters is increasing by 1 filter mass each time a filter is added to the stack. The total downward gravitational force increases doubling, tripling, etc. the net force and the initial acceleration. The air resistance required to bring the filters to terminal velocity must also double, triple, etc. The bottom line then is that the air resistance must be directly proportional to the velocity at which it is falling. Therefore the terminal velocity is directly proportional to the mass of the falling filters.

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What you were doing was conducting an experiment. That is the foundation of science. Congratulations!I assume that you were using conical coffee filters which were all the same, and you varied the weight (or mass) in each experiment by adding a small mass to the apex of the cone. You dropped them all point-down so their descent was stable, and timed the descent over a fixed distance. The cone would accelerate until it reached terminal velocity, when drag = weight.Aerodynamic formulae usually state that aerodynamic force is proportional to the squareof speed but your results seem to show that the force is directly proportional to speed: observed speed is found to be proportional to weight , which is equal to drag ; so drag is proportional to speed .How do you account for this? How accurate are your measurements? Are your results still strictly proportional when you take possible error into account? How did you determine that the coffee filter had reached terminal velocity? Answer these questions and you are doing science!

I suggest, as an educational experience, that you continue the experiment with increasingly massive loads. You will likely find that the proportionality does not continue forever. The linear relation you measured is an approximation to a more complicated one – but valid for low masses and velocities.

The answer to “why” is also a complicated. Terminal velocities of objects falling in air depend on many factors – shape, object density, air density, etc. These determine the upward force of friction (“drag”) which determines termi al velocity. Coffee filters are difficult objects for which to control shape and mass over a large range of mass.

You increased the mass of the coffee filters, so that requires an increased drag to balance gravity at terminal velocity. Assuming you haven’t screwed around with the filters, their drag coefficient and cross sectional area haven’t changed so the only way to get a larger drag is to increase the velocity. Those are the only three things drag depends on. Well, density of the surrounding fluid as well, but you’d have to go into a pressure chamber to vary that.

As Harry Ellis points out, though, that only is proportional at low velocities. Drag is super complicated. In some situations, it is approximately linear in the velocity and in others approximately quadratic. If you get into the quadratic regime, you won’t have a linear relationship any more.

In reality, these are best thought of as the first two terms in a series approximation to the drag force. When v is small, v^2 is really really small so the linear term dominates. Increase v and eventually the v^2 term is bigger and it dominates.