Calculating the velocity of a head-on impact between two suns

[ShaunPereira]

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

 

[Orodruin]

You need to take into account that both stars move.

 

[ShaunPereira]

Yes that is why I have mentioned my attempt at looking at the velocity as relative as both planets move towards each other with the same velocity but I haven't been able to develop the mathematics for it and need help with that

 

[PeroK]

Yes that is why I have mentioned my attempt at looking at the velocity as relative as both planets move towards each other with the same velocity but I haven't been able to develop the mathematics for it and need help with that

There are a couple of ways to do this. Both stars are subject to the same force, which means that they both have the same acceleration. The relative acceleration, therefore, is double the acceleration of each star. Note also that the if the distance between the stars is $r$, then the relative/separation speed of the stars is $\frac{dr}{dt}$ and this is not the speed of each individual star.

Another approach is to think about the motion of each star relative to the centre of mass of the system.

A more sophisticated approach is to use the concept of the reduced mass of the system. You could do an internet search for reduced mass.

 

[ShaunPereira]

There are a couple of ways to do this. Both stars are subject to the same force, which means that they both have the same acceleration. The relative acceleration, therefore, is double the acceleration of each star. Note also that the if the distance between the stars is $r$, then the relative/separation speed of the stars is $\frac{dr}{dt}$ and this is not the speed of each individual star.

Another approach is to think about the motion of each star relative to the centre of mass of the system.

A more sophisticated approach is to use the concept of the reduced mass of the system. You could do an internet search for reduced mass.

Thank you so much
I understood my mistake of not taking the relative acceleration there which was 2a and that the velocity we thus obtain is the relative velocity. Arrived at the right answer.

Reduced mass is a concept I am familiar with(a little) . I have used it sometimes for problems like where two masses connected by a spring oscillate. Never knew that I could use it in a problem like this too.

However I didn't get your approach regarding the centre of mass. All I know is that the centre of mass doesn't move ( in this case) and that the velocity obtained would be the actual velocity in which I am interested and not relative to the other body but I don't know what to take the acceleration as and with what limits should I integrate the acceleration with respect to displacement to get v

 

[PeroK]

However I didn't get your approach regarding the centre of mass. All I know is that the centre of mass doesn't move ( in this case) and that the velocity obtained would be the actual velocity in which I am interested and not relative to the other body but I don't know what to take the acceleration as and with what limits should I integrate the acceleration with respect to displacement to get v

If $r$ is the distance of one star from the centre of mass, which is a fixed point, then we have:$$\frac{d^2r}{dt^2} = -\frac{GM}{(2r)^2}$$

 

[ShaunPereira]

Got it finally! Thank you very much