Calculating Projectile Trajectories with Gravitational and Electrostatic Forces

[BOYLANATOR]

I am an undergraduate physics student and found myself thinking about a scenario where two positively charged projectiles are fired at an angle to the horizontal at a velocity with a certain separation. So they will be subject to a gravitational force and perpendicular to this, an electrostatic force.
The distance dependent acceleration is something I have not covered before and am unsure as to how to calculate where the projectiles will land.

Lets give the projectiles a mass of 1kg and a charge of +1mC. Say they are fired at 45degrees to the horizontal with a velocity of 40m/s. They are fired in the same direction at the same time but separated by 1m. If we say this happens on a co-ordinate system, we can say they are fired from +/- 0.5 x, and parallel to the y axis.

I have worked out that the time of flight is 5.77s and that the y co-ordinates of the impact points are +/- 160.2m.

How can I set up the calculus to solve for x?

 

[haruspex]

The gravitational force is vertical. By symmetry, the electrostatic force is always horizontal, in the x direction. If you're ignoring air resistance, there's no interaction between these. So you can treat the particles as repelling, and moving, purely in the x direction for the known flight time.
Again by symmetry, the repulsion on each ~ 1/x^2:
x'' ~ x^-2
Can you solve that?

 

[BOYLANATOR]

Yes but neither the acceleration or the separation are constant so I am a bit confused.

 

[haruspex]

Quite so. The acceleration (x direction) is as I indicated, is inversely proportional to the square of the separation (x direction). Can you solve that differential equation?

 

[BOYLANATOR]

Sorry, yes you did show that.
Ok so I solve and get x=-c.ln(x)+x₀.
Again I am unsure how to solve this.

 

[haruspex]

Hmm .. that doesn't look right.
You should first get the energy equation: (x')^2 + k/x = E (constant)
But solving that is nasty.
You can think of this as the standard free fall in a gravitational field, for the special case of a straight line "orbit". See e.g. http://en.wikipedia.org/wiki/Free_fall#Inverse-square_law_gravitational_field