 #1
ShaunPereira
 40
 4
 Homework Statement:

Two stars each of one solar mass (=2×10 ^
30 kg) are approaching each other for a head on collision. When they are a distance 10^9 km, their speeds are negligible. What is the speed with which they collide ? The radius of each star is 10^4 km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
 Relevant Equations:

F=Gm1m2/r^2
Energy conservation principle
Firstly I would like to start with solving the problem with energy conservation principle which most solutions to the question show.
Gmm/r= 1/2 mv^2 +1/2mv^2 Gmm/2R
Where m= mass of planet
r= initial seperation
v= final velocity.
R= radius of planet
If we calculate to find v be about 2.6× 10^6 m/s
So far so good
The problem I encounter is when I try to use integration and integrate the infinitesmal distances over which acceleration changes to find final velocity
a= vdv/ds
a ds = v dv
a as a function of distance is Gm/r^2
On integrating we get
V^2/2 = GM/r where r extends from initial to final seperation
Comparing this equation and the one we get by using conservation of energy gives us different answers where velocity is divided by a factor of 2 in the former case and is not in the latter which gives me two answers
I have tried hard to think over the problem even assuming the velocity to be relative velocity between the two planets in the case of the integration but I just can't wrap my head around it
A little help would be appreciated
Gmm/r= 1/2 mv^2 +1/2mv^2 Gmm/2R
Where m= mass of planet
r= initial seperation
v= final velocity.
R= radius of planet
If we calculate to find v be about 2.6× 10^6 m/s
So far so good
The problem I encounter is when I try to use integration and integrate the infinitesmal distances over which acceleration changes to find final velocity
a= vdv/ds
a ds = v dv
a as a function of distance is Gm/r^2
On integrating we get
V^2/2 = GM/r where r extends from initial to final seperation
Comparing this equation and the one we get by using conservation of energy gives us different answers where velocity is divided by a factor of 2 in the former case and is not in the latter which gives me two answers
I have tried hard to think over the problem even assuming the velocity to be relative velocity between the two planets in the case of the integration but I just can't wrap my head around it
A little help would be appreciated