What is gravitation? Well, it is the force that attracts all masses, no matter how small, but you probably already knew that. How do you calculate it? Well, here's the equation:

What does this mean? Well, G is the gravitational constant, approximately 6.67408 × 10^{-11}. The M is one of the masses, and m is the other. r is the distance between their center of masses. What does this mean? Well, using the equation F = ma, or a = F/m, to conclude this:

What could we use this for? Well, to find the velocity. Since a × dx = v × dv, that means we can find out an equation for this. Because the distance is "r", we can find that substituting x in for r in the equations won't make a difference. So here is the derivation:

You can work out the units, it comes out right in the end. In case you still don't believe it, it still conforms to logic:

So, your challenge is to derive an equation for time using integration, and I hope to see answers down below!

Now, what could we use this force equation for? Well, for simulating orbits! I've used data from NASA and the p5.js framework to create a simulation of the four innermost planets' orbits. You can find the code here.

That one guy that's always on

What are r_0 and r_f?

Is r_0 the periapsis, and r_f the apoapsis?

Keeping r constant will make this much easier (just using the semi-major axis)

r_{0} is the initial distance, and r_{f} is the final distance. They can be seen as the major and minor radii, but they can also be seen as the distance from the center of the Earth of a falling object.

Awesome challenge Lucas! Remember we must always apply Newton's law of Gravitation knowing that it is inaccurate as shown by Einstein. For all of the challenges we are able to solve at A level (16-18 years old) the Newtonian way of treating gravity is a more then sufficient approximation. The following is only loosely to do with the challenge (ignore me if you haven't come across these ideas before):

Keeping this in mind I just want to clear something up. To properly simulate the orbits the information you are requesting should be worked out by using the sun's energy distribution and working out the orbits using Einstein's field equations. This actually has practical applications, as the orbit of mercury (especially its procession) is inaccurately predicted by Newtonian gravitation.

I just wanted to mention it, because it is awesome how there is evidence for GR that we can investigate!

More on this topic: https://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity

PlanckTime - Amin

Yes, I am aware of this. I just don't know how to process the equations necessary without slowing down the program. Plus, this was only to use Newton's equations first, then lead up to Einstein's in the next post, hopefully creating a program that accurately shows all of the orbits using General Relativistic gravity and outside forces without any lag.