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When a body with a mass of 1 kg moves at constant velocity of 1 m/s and collide (elastic collision) with a body with the same mass that's at rest, we know from conservation of momentum and conservation of energy that the first body will stop and the other body will start moving at 1 m/s.

Newton's second law states that $F=m\frac{dv}{dt}$, therefor velocity has to be continuous (otherwise it doesn't have a derivative).

In our collision, the velocity of the second body before the collision is 0 m/s, and later it's 1 m/s. From intermediate value theorem we get that at some point in time, the velocity is 0.5 m/s.

The two bodies are mass-equal, therefore from conservation of momentum we get that at that point in time, the velocity of the first body is also 0.5 m/s.

Calculating the total kinetic energy will give us $\sum E_k=0.25\text{ J}$

before the collision, the total kinetic energy is $\sum E_k=0.5\text{ J}$

That seems to be a contradiction to conservation of energy.

Am I missing anything?

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    $\begingroup$ We usually treat collisions as instantaneous to simplify things, but like you point out, a more realistic model would treat the velocities as continuous. For this case, it is helpful to think of the two bodies as groups of atoms connected by springs. During the time period when the two bodies are colliding, they are compressed and the imaginary springs store the energy as potential energy. As you can imagine, the springs rebound and release energy quickly afterwards. For elastic collisions 100% of the energy is converted back into kinetic energy. $\endgroup$ Mar 15 at 19:38
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    $\begingroup$ This is an answer. In fact, it is the answer. So it is not a comment. $\endgroup$
    – my2cts
    Mar 15 at 20:25

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There is no contradiction between conservation of momentum and energy and Newton’s 2nd law. Your analysis has one small mistake that makes the conclusion wrong.

Specifically, the conservation of energy does not imply the conservation of kinetic energy. Kinetic energy is not conserved in general. So finding a change in kinetic energy is insufficient to show non-conservation of energy. Instead you would need to show a change in total energy. As others have indicated, in this case the other energy is elastic potential energy. The total energy is conserved.

Now, you might say that kinetic energy is not conserved in general but that kinetic energy is conserved in an elastic collision. So you might say that while the above objection would be correct in general, it doesn’t hold in this specific case.

However, this is a misunderstanding of the meaning of an elastic collision. An elastic collision means that the kinetic energy immediately before the collision is equal to the kinetic energy immediately after the collision. It says nothing about the kinetic energy during the collision. So again, kinetic energy going to elastic potential energy doesn’t contradict the elastic collision constraint either.

Finally, in comments you argued that no elastic potential energy can be stored in an infinitesimal body. This is simply an incorrect claim as a mathematical model. If you write down the actual math you will find that the amount of elastic potential energy stored when $v=0.5 \ \mathrm{m/s}$ is constant and independent of the size of the object, regardless of how small.

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we know from conservation of momentum and comservation of energy that the first body will stop and the other body will start moving at 1 m/s.

Kinetic energy is preserved only if the collision is perfectly elastic. This means that while the two objects are in contact, part of energy will be stored as elastic potential energy which is later converted back to kinetic energy.

therefor from conservation of momentum

Momentum is conserved only if there are no external forces acting on the system of particles. We also say that the momentum is conserved if external forces are much smaller in magnitude than the collision force that forms between the particles, but this is only approximation. If you want to know more, check what impulse is in context of forces.

That seems to be a contradiction to conservation of energy.

Am I missing anything?

Yes, during (elastic) collision there is a portion of energy stored as elastic potential energy.

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  • $\begingroup$ According to what I found in Khan academy's website, elastic potential energy exists only when a force is trying to deform the body. But if the bodys in my case are infinitesimaly small (as in simple Newtonian mechanics), then there's no such a thing as "deforming" them. (I know infinitesimaly small bodys don't really exist, but I'm asking about Newtonian mehcanics, not about our real univers. You can think of my question as a question about a mathematical model) $\endgroup$
    – Ben
    Mar 15 at 20:19
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    $\begingroup$ @Ben When you study collisions in newtonian mechanics there is no assumption about the point-like character of the objects. You can consider them point-like but then there is no intermediate phase between initial and final state. Either you consider an idealized case or a more realistic. You can't have half of both. $\endgroup$
    – nasu
    Mar 16 at 0:21
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The solution is that energy is stored in the deformation of the objects. To maintain your argument you could assume incompressibility. However, in this limit the time interval during which both objects move simultaneously goes to zero. In this limit your argument therefore no longer holds. This limit would also require infinitesimal size in order to be compatible with special relativity.

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    $\begingroup$ Time goes to zero - so what? conservations of energy and momentum and Newton's second law still hold, so everything is the same. $\endgroup$
    – Ben
    Mar 15 at 20:55
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    $\begingroup$ @Ben if the interaction time is zero, the velocities are discontinuous, and so your mean value argument is inapplicable. $\endgroup$
    – Gilbert
    Mar 16 at 2:59
  • $\begingroup$ How can velocity be dicontinuous? I showed from Newton's second law that it has to be continuous (see my original question above). $\endgroup$
    – Ben
    Mar 16 at 11:56
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    $\begingroup$ @ben It's up to you. If you consider the objects compressible, then potential energy needs to be taken into account. If you do not accept this, the bodies must be incompressible but this implies that the velocities are discontinuous and the force is infinite. $\endgroup$
    – my2cts
    Mar 16 at 20:37
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The total energy of the system must be constant, and before and after the collision it is the kinetic energy of one of the bodies.

But during the very small $\delta t$ of what we call contact, the potential electrostatic energy between the atoms of the surface can not be neglected. It is exactly this potential energy the source of the force of the impact on the objects.

So, during $\delta t$ the kinetic energy is smaller, and the sum of potential and kinetic energy keeps constant.

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