## Saturday, July 12, 2008

### Exponential acceleration

I want to do some reading today, so this is just a short post. It starts with a bit of physics, but it is actually about an the exponent. A few days ago, a strange thought came into my mind. This thought was very simple: What is the fastest way to move, from mathematical perspective? The answer was obvious - the fastest way is to move with exponential acceleration, e^x. However, this thought continued to bug me, so I started to think about this problem in more details. It seems boring, but it brought me an surprising result.

Firstly, what is so special about e? Is it correct to say that exponential acceleration is the fastest one? After all, 10^x>e^x. However 10^x=e^(xln10)=10e^x. While it is bigger, it is just e^x multiplied by a fixed number.

Now to the more interesting part. The function e^x has the interesting property that its integral it also equal to e^x. This means that the speed, which is the integral of the acceleration is also e^x. So, how many time will it take for the speed to become greater than the speed of light, given exponential acceleration? To calculate it, we will first suppose that the speed is given in meters per second. The speed of light is then about 3*10^8=300000000 meters per second. But:

$e^{20}>485165195$

And this is a significantly large number. What this means is that it will take only 20 seconds to
get to speed higher than the speed of light (provided relativity isn't working). If we would continue this way, after 37 seconds our speed would be over one light year per second, and after 5 seconds more it would be over 100 light years per second.

We all now that for any n:
$lim_{x->\inft } \frac{x^n}{e^x}=0$

Which can be described as "the exponent goes to infinity faster than x^n, for all n". This description is not mathematical, (if it is faster how much kmh is it doing?) but the simple calculation I did above showes well what it means. The exponent is a totally wild function...