how can I solve this question?

[esradw]

A comet is released from rest at a distance r max from the sun
(angular momentum = 0 ) How long the comet takes to reach the sun? ( it says use the technique that t = integral ( dx/x(dot) where we find the Xdot from T = E-U )

I understand that since angular mom=0 c=0 so c=rmax(1-e) e=1 a (semimajor)=rmax/2 and c=rmin(1+e) for c to be equal to zero rmin ( semiminor must go to zero. Right ? but But I still can not find the right answer :(( ( t= (pi/2*root(2GMs))*(r max)^3/2 )

As I thought there is only one force acting on the comet when I do F=md^2r/dt^2 I find rdot and with the help of the hint I find t = rmax^2/2GMs but the solution is t= (pi/2*root(2Gms)) *(rmax)^3/2 which is kepler`s IIIth law. How can I find this solution ? Do you have any idea ?

I`d appreciate that anyone could help me, I need to return the homework tomorrow
Thanks

[OfficeShredder]

Forget about Kepler's laws. If the angular momentum is zero, it means the comet is falling straight towards the sun. So it's just a force of GMm/r^2 pulling on the comet from r=r max to 0, or an acceleration of GM/r^2.

Does that help?

[HallsofIvy]

dr^2/dt^2= -GM/r^2

Let v= dr/dt so that dv/dt= -GM/r^2.

But, by the chain rule, dv/dt= (dv/dr)(dr/dt)= v dv/dr so the equation becomes v dv/dr= -GM/r^2. v dv= (-GM/r^2)dr or (1/2)v^2= GM/r + C (which is "conservation of energy" in disguise). Since v= 0 when r= R, the initial distance, C= -GM/R and v^2= 2GM(1/r- 1/R). v= dr/dt so this becomes
dr/dt= sqrt(2GM(1/r- 1/R)).

dr/sqrt(1/r- 1/R)= sqrt(Rr/(R-r))dr= sqrt(2GM)dt.

Can you integrate that?