# In a collision shouldn't objects of different mass have same acceleration?

Suppose two objects of different mass, A and B, collide with each other. Now, during the time of collision, they both apply forces on each other according to Newton's 3rd law. Therefore, their accelerations won't be equal.

Here, my confusion is that, while they are colliding, they are both touching each other. Hence, the kinetic properties of one object must coincide with those of the other one as well (since you cannot keep two objects with different accelerations and velocities in touch for a non-zero amount of time). Therefore, their acceleration must be equal to each other at any instant. Now, this seems to contradict Newton's 3rd law. Where is the problem?

PS: I am talking about elastic collisions.

• Consider what "elastic collision" means. Commented Mar 14, 2021 at 1:33

If you model the colliding objects as rigid bodies, then the collision will take zero time. Since the collision has zero duration, there is no issue with the objects having different accelerations. While this is often a reasonable approximation to make to simplify calculations, it doesn't exactly explain really going on here, so we look at what happens if we take away this assumption. If you do not model the colliding objects as rigid bodies, then you must account for how the objects deform during the collision. Since the objects are deforming, their centres of mass can have different accelerations without the objects losing contact, because the sizes of the objects will not be constant, i.e. the distance between the centres of mass can change while the surfaces of the objects stay touching, like compressing a pair of springs.

The elastic collision takes place as such:

Firstly the objects with different speeds approach each other. Once they come in contact with each other, the approaching body pushes its way through the body in front of it. In this way, both experience deformation. During this period, both bodies suffering equal forces (third law of Newtonian motion) experience same impulse. The faster one is forced to retard and the slower one is forced to accelerate. At the time instant of maximum deformation, the bodies would experience zero relative velocity which clearly implies that both are moving at the same velocity. Assuming no heat loss, rest of the mechanical energy is stored as the elastic potential energy in the deformation. After that the bodies start of elongate and restore their original shape (elastic property of the body). Thus the body in front, while expanding itself, pushes the body at the back and the body in the back pushes the front body in the same way. Hence they again experience forces in the same direction as they experienced before maximum deformation. Hence, the body at the back gets retarded even more and the body at front accelerates even more. This force pair ends working when the bodies loose contact.

So, through the above description I intend to convey that the collision is a property of "non-point-sized-bodies" though for ideal cases we consider them as one. Thus the kinetic properties of the whole bodies can't be same even when in contact. Only the places where they are in contact, they share same kinetic properties. But when they are at maximum deformation, you can expect them to show same kinetic properties.

Here, my confusion is that, while they are colliding, they are both touching each other.

This is correct

Hence, the kinetic properties of one object must coincide with those of the other one as well

This is also correct

(since you cannot keep two objects with different accelerations and velocities in touch for a non-zero amount of time).

This is not correct, with a proviso. It's their change of momentum from before and after the impact that is equal. Moreover, for the duration of the impact, the rate of change of momentum of one is the opposite of the rate of change of momentum of the other.

Therefore, their acceleration must be equal to each other at any instant.

This doesn't follow becaise it's not acceleration that is yhe kinetic property we should focys on bit momentum and which mass x velocity.

Now, this seems to contradict Newton's 3rd law. Where is the problem?

As I've indicated above.

The proviso is that Aristototle in antiquity debated the reality of atoms. At the time, one model of atoms were as hard and impermeable objects. He pointed out that this could not be the case as otherwise they simply couldn't touch. Modern day physics has backed up his notion with atoms being seen as fields.

During collision ha no requirement of equal kinetic energy or acceleration for the bodies undergoing collision.

There can be collisions where one body is at rest while other body is at some velocity hits collides. In fact this is very common type of two body collision seen around.

Touching each other is different from hugging each other. I mean just imagine you standing on the roadside and your friend who is riding a bike, gives a high-five while moving. There is absolutely no reason for two bodies two be at the same velocity/acceleration for collisions to happen.

Contact Collisions in Classical Mechanics is a INSTANTANEOUS PROCESS.

(Since you cannot keep two objects with different accelerations and velocities in touch for a non-zero amount of time)

Pretty sure you can. I've seen it in real life. If I throw a ball at a wall, the ball suffers an acceleration back at me. Its velocity goes from +something to zero and then to -some other value. It was in contact with the wall for the whole time, which is not zero (it's not instantaneous). The wall experienced a very different set of accelerations and velocity changes throughout.