[OT] Spheres in collision

[Aubrey McIntosh, Ph.D.]

Like any of you ever thought my posts were on topic ...

(crossposted, trim) I'm sure this will interest the s.e.d regulars and
several can solve it, and s.s.science & rec.models.rockets are my best
guess for where it actually is on topic. Any other groups likely to be
appropriate? This is related to my post about eMachineShop in other
groups this week.

Given two spheres of mass \r and density \rho, each has a mass of \rho V
= 4/3 \pi \rho r^3.

Separate them, center to center, by 3r. That is, surface to surface
separation of r, then the gravitational force between them is

F = G m m / (9r^2)

Using F = ma and waiving hands wildly while students sleep ...

a = -G m / (9r^2)
a = -4/27 \pi \rho G r

Using iridium (for small spheres), \rho = 22,160 kg/m^3
G=6.673E-11 m^3 / (kg s^2)
a = -688E-9 m/s^2 r

Notice, a is reletive to center of mass, not sphere to sphere,
explaining a factor of 2 that likes to hide.

Solving s=1/2at^2 and bounding t by the initial value of a:

t = 1,205s or 20 min. Notice, it doesn't matter what r is.
......
Now, call the original position \s_0 at time t=0 and place the origin at
the center of mass. At time t, the position is s, and the separation
between centers will be 2s. There is surely more elegant notation.

a = 4/3 \pi G \rho r^3 / s^2

s(t) = \int \int (4/3 \pi G r^3) / (2s)^2 dt dt

Maybe I'm slow this morning, but I don't see the solution to this. Can
someone with untarnished calculus or mathematica give a solution to
this, for arbitrary s_0? Is there some aerospace engineering text where
this is treated as an example or homework problem?

 

[John Larkin]

On 14 May 2005 15:11:59 -0700, "lunarlosREMOVE2EMAIL@juno.com"
<lunarlos@juno.com> wrote

I wish I could be more helpful but it has been many years since I left
college and I just can't think as sharply as I once did ... time, time
dumbs us all to death eventually.

But whereas brute computational abilities may decline with age and
disuse, shear sneakiness and guile more than compensate. In other
words, you can always hire kids to to the heavy lifting.

John

 

[Anno Siegel]

Aubrey McIntosh, Ph.D. <newsposter@spam.vima.austin.tx.us> wrote in sci.electronics.design:

[...]

Given two spheres of mass \r and density \rho, each has a mass of \rho V
= 4/3 \pi \rho r^3.

[...]

s(t) = \int \int (4/3 \pi G r^3) / (2s)^2 dt dt

Maybe I'm slow this morning, but I don't see the solution to this. Can
someone with untarnished calculus or mathematica give a solution to
this, for arbitrary s_0? Is there some aerospace engineering text where
this is treated as an example or homework problem?

By differentiation wrt. to t (twice) it becomes an ordinary differential
equation. In Mathematica notation:

s''[t] == 4/3*Pi*G*r^3/(2s[t])^2

or

deq = s''[t]*s[t]^2 == Pi*G*r^3/3

Mathematica solves it, but the result isn't pretty (see below). You'll
note the solution is in terms of two nasty transcendental equations. I
haven't followed this further.

Anno

---------------------------------------------------------------------------
In[10]:= DSolve[ deq, s[t], t]

Solve::tdep: The equations appear to involve the variables to be solved for in
an essentially non-algebraic way.

Solve::tdep: The equations appear to involve the variables to be solved for in
an essentially non-algebraic way.

3
Out[10]= {Solve[C[2] - Sqrt[3] ((G Pi r

3
-2 G Pi r + C[1] s[t]

+
s[t]

3
2 (-(G Pi r ) + C[1] s[t]) 3/2

+
Sqrt[C[1]]

3
-2 G Pi r + C[1] s[t]
s[t] Sqrt[----------------------]
s[t]

,
C[1]

3

((G Pi r

3
-2 G Pi r + C[1] s[t]

+
s[t]

3
2 (-(G Pi r ) + C[1] s[t]) 3/2

+
Sqrt[C[1]]

3
-2 G Pi r + C[1] s[t]
s[t] Sqrt[----------------------]
s[t]

}
C[1]