Gravitational attraction and acceleration over time.

[MrJingles]

Suppose I have 2 bodies, M1 and M2. Assuming that no other forces are acting on them aside from gravity, what is the nature of their acceleration over time? As they approach each other, and their distance decreases, does their acceleration also increase? If so, what is the rate at which this occurs, and with what equation might one find the acceleration or velocity at a specific time, t?

Does it have something to do with F=d(ma)/dt and F(G)?

Also, how might one calculate the gravitational attraction between more than two bodies?
How might one determine the acceleration of each body at a given time as they approach each other?

I only have a limited understanding of calculus (calc 1) and vectors, but do not hesitate to explain in full. Thanks for your time.Oh yes- and do orbits come into play here?

 

[MrJingles]

To avoid an absurdly complex problem, let's assume that all these objects magically came into existence without any initial velocity or momentum.

 

[KKW]

The force at which the bodies attract each other is F= GM1M2/d^2, where G is a constant. So when they approach each other the d, distance btwn them, decreases, causing the attraction to increase and the bodies would accelerate towards wach other. Think of g=9.81m/s^2. That is an accelation.
Orbits don't come into play- the objects would travel in a straight line. Introduce a third object and things would bcome more complicated and maybe orbits would appear.

 

[Ulysees]

I had exactly the same questions a long time ago.

what is the nature of their acceleration over time?

Acceleration is proportional to force, and force proportional to the inverse square of the distance between them, therefore as they get closer acceleration increases. You can express acceleration as a function of time analytically for 2 objects, but the maths is not trivial and cannot deal with more than 2 objects. So the general approach is numerical integration:

You take small steps in space based on the speed at each moment, and small speed-steps based on the acceleration at each moment. That is, for N objects you write it like this:

dv1/dt = acceleration a1 = f1( distances between objects )

dr1/dt = velocity v1

dv2/dt = acceleration a2 = f2( distances between objects )

dr2/dt = velocity v2

...
dvN/dt = acceleration aN = fN( distances between objects )

drN/dt = velocity vN

Eg for object 1, whose state is made up of v1 and r1, the steps are like this:
step in space dr1 = v1 * dt
step in speed dv1 = a1 * dt
a1 = f1( r1-r2, r1-r3, ... r1-rN )

Function f1() is the vector sum of the gravitational forces on object 1 due to all other objects (2, 3, .., N), divided by M1 to get the acceleration:

f1(r1,r2,...rN) = 1/M1 * ( G * M1 * M2 / |r1-r2|^2 * (r2-r1)/|r2-r1| + ... )

This expression (r2-r1)/|r2-r1| is just used to get the direction of the vector of force on object 1 due to object 2. The rest is Newton's law of gravity, applied for each pair of objects in turn (only pair 1-2 shown).