Newton's Third law for two objects in an isolated system with only one internal force acting between them

Question

I have been thinking about Newton's Third law from the past three days and am not sure if i completely understand it.

I need some help answering/describing this situation i had been thinking about. Answers to this will tremendously help me in understanding the third law.

Say in an isolated system with no external forces exists two initially seperated, non-deforming objects A and B and the only force acting between the two objects is the force of gravity (i.e none of the other three fundamental forces are in play here). Initially both the objects are in rest. Due to gravity they will obviously collide after some time.

My question is, what will happen after collision? Isn't it impossible for the objects to stay collide and bounce away from each other? Won't the action reaction forces cancel each other out at collision in this situation?

please someone correctly explain in depth what will happen as time passes by in the above system, to me.

Answer 1

Newton's Third law for two objects in an isolated system with only one internal force acting between them
According to Newton's third law this is impossible there must be two forces, the force on object $A$ due to object $B$ and an equal but opposite force on object $B$ due to object $A$.

Won't the action reaction forces cancel each other out at collision in this situation?
This is a common mistake and one must note that the action and reaction forces act on different objects so there can be no cancellation of forces.

My question is, what will happen after collision?
Anything you decide on as long as the Laws of Physics are not violated.

If the collision was elastic (kinetic energy is conserved) then they might rebound and return to their original positions and then start the sequence all over again, and again, . . . .
In the real world not really consistent with the statement non-deforming objects.

They might stick together with the initial gravitational potential energy of the system being used to do work cause permanent deformation of the objects, converted into heat etc, . . . .
Not really consistent with the statement the only force acting between the two objects is the force of gravity (i.e none of the other three fundamental forces are in play here) which is also true of my first example.

Thus the scenario that you have used is not a real world one and needs to be amended to answer the question What happens next?

Answer 2

Although Newton's Second Law is for a single particle, $\vec{F}_{\rm net} = m \vec{a}$, we can sum many of these for systems of interacting particles (applying Newton's Third Law along the way) to obtain an effective equation for the system: $$ \vec{F}_{\rm net,\, ext} = M_{\rm sys} \frac{d \vec{v}_{\rm com}}{dt} $$ where $\vec{v}_{\rm com} = \frac{d \vec{r}_{\rm com}}{dt}$, with $\vec{r}_{\rm com}$ the center-of-mass position of the system of particles, $M_{\rm sys}$ the sum of their masses, and $\vec{F}_{\rm net,\, ext}$ the sum of the external forces on the system. See my answer to this question, for example.

So, for your question, where $\vec{F}_{\rm net, \, ext} = 0$, the main thing we know about these two particles is that their center of mass postion will remain at fixed velocity for all time: $$ \vec{v}_{\rm com} = \overrightarrow{\rm const} $$
To describe their exact dyanamics, you would need to think of them not as particles, but "balls" with some extent (rigid collections of particles), and write down the rules for their interaction during collision, e.g., elastic collision of spheres. Instead of using Newton's Third Law to relate their interaction forces during these collisions, you should just use conservation of momentum (which is precisely derived from Newton's Second and Third laws; again, see my answer linked above). That will relate the final velocity vectors to the initial velocity vectors in each collision. See this wikipedia entry on an elastic collision in two dimensions.

The precise motion of the two will depend on their initial conditions, but, as long as the (magnitude of the) difference in their initial velocities is less than the gravitational escape speed, your description of gravitational attraction and collision, on repeat, sounds right.

Answer 3

The action reaction pair are A pulling with gravity on B and B pulling with gravity on A. Any force must be between two objects. The force pulls the same on both objects, thus A and B feel the same amount of force from the gravitational force between them, but in opposite directions.

When A and B come in contact, surfaces touch. Whenever two surfaces make contact, a normal force (force perpendicular to the surface) forms to prevent the surfaces from passing through each other. This is not because of gravity. This is only because the objects touch each other. Gravity pulls objects together. Normal force pushes objects apart. Normal force will be as strong as it has to be to keep the surfaces from breaking, up to a limit. If the surfaces are elastic, they will bend during the time of the collision. Like a spring, the surfaces will then push back out to send the objects flying apart. If the surfaces are not flexible and the normal force's limit cannot stop the motion in time, then the surfaces will be permanently bent or perhaps significantly damaged. The normal force between A and B may be strong enough to eventually stop the motion but not strong enough to push them back apart. In this case the objects will stay together.