Hi, I'm trying to work out the following equation that I have made for myself, based on the following information:
An object of mass m is travelling from a point r_0. The object has two forces acting on it, both of which are inversely proportional to the square of the distance from r_0.
I know this means that the ratio between the two forces is constant, and I have found the acceleration by using F=ma, with F being the difference between the two forces, divided by r^2.
How do I go about solving this equation for velocity at a certain distance r from r_0?
I know that (using simplified notation, lumping all constants together) a = (A-B)/mr^2 and that a = dv/dt, so we have a first order constant coefficient differential equation. However, it is the r^2 term that is causing issues, as this means it can't be solved easily.
Am I missing a trick here?
The Attempt at a Solution
The reason why I wish to solve this is that this equation models the acceleration of a spaceship whose only forces acting upon it are the gravitational pull of the sun, and the radiation force acting on the sails of the spaceship. Both forces decrease like 1/r^2 as the ship moves away from the sun, hence the above equations. I am trying to find at what distance from the sun a particular velocity is reached, if at all. (As I imagine that if the difference between the two forces is not large enough, then this speed will never be attained)
As an aside, it is interesting to note that even if the radiation force is only slightly larger than gravity, then the spacecraft will eventually leave the solar system, and the sun's gravitational effect. But it will stop accelerating at some point, or the acceleration will become so small as to be negligable. But there have been no forces to slow it down, so it will be travelling at a great speed.