Hi, I'm trying to solve Newton Universal law of Gravitation as a differential equation in one dimension with two objects. One mass is fixed at distance r = 0, whilst the other is some initial distance away. I'm using distance rather than displacement and also ignored the proportionality factor to make the formula simpler. So in essence, this is the differential equation I want to solve:
\(\displaystyle a = \frac{d^2r}{dt^2} = -\frac{1}{r^2}\)
However, I've hit a snag when trying to integrate for the 2nd time (see the image file). I have used two substitutions (a and b), and the last line in the file suggests that I will obtain the inverse function that will contain hyperbolic and polynomial terms, i.e. I won't be able to determine the original function of r with respect to t.
Can anyone help?
\(\displaystyle a = \frac{d^2r}{dt^2} = -\frac{1}{r^2}\)
However, I've hit a snag when trying to integrate for the 2nd time (see the image file). I have used two substitutions (a and b), and the last line in the file suggests that I will obtain the inverse function that will contain hyperbolic and polynomial terms, i.e. I won't be able to determine the original function of r with respect to t.
Can anyone help?