The dynamics of a body falling toward a mass M a distance r away from it is given by the equation

d²r/dt² = −GM/r²
 

where G is the gravitational constant.

Suppose the body is initially at rest at a distance of r₀. If the above equation is multiplied by v=(dr/dt) and integrated, taking into account the initial conditions the result is

½v² = GM[(1/r)−(1/r₀)]
thus
dr/dt = (2GM)^1/2[(1/r)−(1/r₀)]^1/2
 

If this equation is divided by r₀ on the left and equivalently muliplied by (r₀)^1/2/(r₀)^3/2) on the right and y substituted for r/r₀ the result is

dy/dt = (2GM/(r₀)³))^1/2[1/y - 1]^1/2
 

A further substitution of 1/y-1=z and y=1/(z+1) gives

dy/dt=-(1/(z+1)²)(dz/dt) = (2GM/(r₀)³))^1/2z^1/2
which leads to
(1/(z+1)²)(1/z^1/2)(dz/dt) = −(2GM/(r₀)³))^1/2
and finally
(2/(z+1)²)(d(z^1/2)/dt) = −(2GM/(r₀)³))^1/2
 

Now let z^1/2=q, and hence z=q². The above equation then takes the form

(2/(q²+1)²)dq/dt = −(2GM/(r₀)³))^1/2
 

Indefinite integration of this equation gives

q/(q²+1) + tan^-1(q) = −(2GM/(r₀)³))^1/2 + C
 

where C is an arbitrary constant.

The variable r ranged for r₀ to 0 so y ranged from 1 to 0. The variable z=1/y-1 then ranged from 0 to ∞ and likewise for q=z^1/2.

For an integration with respect to q from q=0 to q=∞ the first term, q/(q²+1), vanishes at both the upper and lower limits. The second term, tan^-1(q) has the value π/2 for q=∞ and 0 for q=0. The value of the integral is −π/2 because of the inversion of the limits in going from y to q. Let t_f be the time at which r goes to 0. Then

π/2 = (2GM/(r₀)³))^1/2t_f
and hence
t_f = (π/2)((r₀)³/(2GM))^1/2
 

For a comparison consider a particle moving under a constant acceleration g. In time t it travels a distance ½gt². Thus the time t_c required to travel a distance r₀ under constant acceleration is

t_c = (2r₀/g)^1/2
 

The value of g is GM/r₀² so the above expression reduces to

t_c = (2r₀³/GM)^1/2
 

This leads to the interesting result that

t_f/t_c = π/4
 

That is to say, the time which it takes a particle to reach the center of the gravitational attraction is 78.54% of the time it would take to reach it under constant acceleration equal to its initial acceleration.

A Spherical Gas Cloud Without Any Interaction of Particles

If the mass is distributed uniformly throughout a spherical cloud the value of M depends upon r₀. In fact

M = (4/3)πρr₀³
 

where ρ is the mass density.

When this expression is substituted for M in the equation for t_f the result is

t_f = (π/2)(1/(2G(4/3)ρπ))^1/2
 

Thus t_f is independent of r₀. This means all elements of the spherical cloud collapse to the center at exactly the same time! Of course this would be prevented by the build up of material and pressure in the process of collapse.