## November 20, 2010

### Force is proportional to acceleration. What is the constant?

Filed under: internet — takaita @ 11:53
Tags: , ,

Somewhere I bumped on this captcha question: “Force is proportional to acceleration. What is the constant?“. The answer is supposed to be “mass”. But that makes little sense.

To clarify, the phrase “Force is proportional to acceleration” refers to F=m.a, the second of Newton’s laws of motion: “A body of mass m subject to a force F undergoes an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the force and inversely proportional to the mass.” (quoted from that wikipedia article).

Then comes the captcha question: “What is the constant?“. The problem is that there is no constant in that formula.

The wikipedia has an article about mathematical constants, starting with the line that it is a special number. Numbers have no dimension, mass has a dimension (kilogram), and so do the other elements in Newton’s law: force is measured in Newton, acceleration in meters per second per second. No mathematical constant in F=m.a.

A physical constant then. Again no, because it is described as “a physical quantity that is generally believed to be both universal in nature and constant in time“. This does not apply to anything in F=m.a.

There is another wikipedia article: Constant (mathematics), which states “In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable (i.e. variable quantity), which is a symbol that stands for a value that may vary.” Does this make sense in the context of this captcha? Almost, but not completely. Force is proportional to both mass and acceleration. It does not imply that mass must be constant to make the formula. In a discussion elsewhere someone came up with a rocket which is burning fuel in order to gain speed. By burning fuel it loses mass. However this does not mean that in this case the force is no longer proportional to acceleration. F=m.a is also valid for this rocket with changing mass. Mass does not need to be constant to make the formula work. The rocket does not violate Newton’s law.

Then shouldn’t the question rather ask for “constant” (adjective) in stead of “the constant” (noun)? That would imply that the first phrase not just hints at (or describes) Newton’s law, but describes a specific situation where Newton’s law is active. Something like “over a period of time, force is proportional to acceleration”, followed by the question “what remains constant?”. That would make sense. But it is not what the captcha says.

As a side note: I started a discussion about this on reddit. Nobody agreed with me, so anyone else is welcome to disagree. Please stay a little polite. The discussion on reddit got a bit out of hand in the sense that I got personal messages telling me a.o. that I was an all-caps fucking imbecile. It was probably just a troll, but the internet seems to make people angry for no reason sometimes.