- #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