Collision of Unequal Mass Particles in a Gravitational Field

[gianeshwar]

Dear friends!
If there are two particles of unequal masses in a system on which there are no external forces.When let free they collide under the influence of gravitation.
Should they always collide at CM.
If the system were under an external force of uniform gravitational field will the collision occur at CM only.
Thanks!

 

[andrewkirk]

Draw a picture of the two particles and mark by dots the locations of their centres of mass. Draw a line between the two dots. Then draw two vector arrows, showing the direction of the gravitational force on each of the two particles.

Then you need to clearly specify what you mean by 'collide at CM'. For almost all particles, the CM is inside the particle, so it is impossible for the CMs of the two particles to touch one another. Which definition you choose will affect the answer to the second part of the question.

 

[gianeshwar]

Thanks andrewkirk! Yes I understand their centre of mass cannot touch due to their sizes as they are assumed rigid bodies.
If I assue them to be point particles then what happens?

 

[mathman]

If there are no other things around, the center of mass of an n-body system does not change, whatever the individual objects do.

 

[andrewkirk]

With two point particles and no other bodies, the forces on the two bodies are along the line that connects the two particles, so they move towards one another. Is there any reason for either of them to deviate from that line? If not, they will eventually collide.

If there's a uniform gravitational field, imagine it in an x-y plane, first where the two points have the same y coordinates and the grav field points in the positive y direction. Then you can consider the vertical and horizontal components of velocity separately, because the axes are orthogonal and do not change direction. The horizontal components are the same as in the first case. So if in the first case the two particles collide at time T, what happens in the second, assuming the only difference between the two cases is the gravitational field?

More generally, the grav field may not be orthogonal to the line between the two particles. In that case, it imparts an additional horizontal acceleration to both particles, as well as the vertical accelerations. But since that horizontal acceleration is the same for both, it doesn't affect when they collide.

 

[sophiecentaur]

Should they always collide at CM.

The geometry of the situation means that has to be correct if the bodies are points. When they are together, they are at the CM. Where else could the CM be? It has to be between them and can't be outside the system. If one body is much bigger than the other then they won't get close enough and the CM can be below the surface of one of them.

 

[gianeshwar]

Thanks mathman,andrewskirk and sophicentaur!