One of the things I find mildly amusing is the way that physics and maths words get taken up into everyday vocabulary, where they take on a slightly different meaning from the original. The word ‘random’ seems to be a favourite in NZ at present, as in "I bumped into this random guy and he said this random thing". Others include ‘infinite’, which means merely very big "The All Blacks were infinitely better than the Wallabies"; and ‘exponential’, which means increasing rapidly "The NZ dollar is increasing exponentially".
The science word I’m trying to get into everyday language is ‘flocculate’. As in "Teenagers flocculate via text messaging". Sounds awfully rude, but isn’t. Go look it up. So far I am failing miserably.
Anyway, back to ‘exponential’. It is colloquially used to mean something that increases rapidly, but its mathematical definition is much more precise. Examples of exponential growth and, particularly, exponential decay are plentiful in physics.
A process is exponential if it changes at a rate proportional to how much there is of it. A deposit in a bank account is a good example. If I put one hundred dollars into an account at five percent interest a year, at the end of the first year there is one hundred and five dollars, that is one hundred times 1.05. After the second year, however, there is 110.25 dollars. That’s the hundred and five times 1.05. The more money there is in the account, the more interest I get. The amount of interest is in direct proportion to the amount there. That’s exponential growth.
A physics example of exponential decay is radioactive decay. If I have a radioactive substance with say 1000 atoms in it (not very many), it may, after a year, have 800 atoms left. That is, 200 have decayed in the first year, or a decay rate of 20% a year. After the second year, there will be 640 left (which is 800 times 0.8), and after the third year there will be 512 (640 times 0.8). The number decaying every year gets less, because the number of radioactive atoms gets less. That’s why we talk about a half-life for a radioactive element, not a full-life. We can define a time when the half the atoms will have decayed, but not a time when they all will have decayed.
Not everything that grows or decays is exponential. For example, the way a ball gains velocity when it is dropped is not exponential. The rate at which it picks up velocity is constant – at about ten metres per second, every second – it isn’t dependent on how quickly the ball is moving (until it gets very fast and air resistance becomes an issue.)
So, colloquially, we might say a ball gains speed exponentially (meaning very fast), but physically, that would be wrong. That is why I do get slightly annoyed when I hear such random comments using ‘exponential’ in an infinitely wrong manner.