Proton Acceleration: Newton's Law & Coulomb's Law

[skepticwulf]

Using the Coulomb's law, the force between two protons inside a nucleus is about 14.4 N
Isn't that a colossal force for such a tiny particle?. I guess that shows how strong nuclear force must be to overcome this.

When I divide this however to the mass of proton, the acceleration due to Newton's law comes up as

=8.6 x 10^27 m/sec^2

That's far far beyond the speed of light for 1 second of acceleration, even for a fraction of second.

Doesn't that contradict with the special reality that says nothing goes beyond the speed of light??

 

[Nugatory]

Newton's law comes up as

=8.6 x 10^27 m/sec^2

That's far far beyond the speed of light for 1 second of acceleration, even for a fraction of second.

Doesn't that contradict with the special reality that says nothing goes beyond the speed of light??

No. Newton's law is $F=\frac{dp}{dt}$ where $p$ is the momentum. (The $F=ma$ that you're probably thinking of is for the special case of constant mass and speeds that are small compared with the speed of light). Use this, and remember that in relativity the momentum is defined as $\frac{mv}{\sqrt{1-v^2}}$, and you won't get that faster than light speed.

Of course none of this is specific to the case of the nearby protons. Let any force, no matter how weak, act on a body for long enough and it will accelerate to speeds that require relativistic instead of classical treatments.

 

[Meir Achuz]

I think you have lost many powers of ten in your calculation.

 

[jtbell]

That's far far beyond the speed of light for 1 second of acceleration, even for a fraction of second.

If you make the time small enough, you can make the final speed as small as you want, no matter how large the acceleration is. But as Nugatory noted, you really need to do the calculation using relativistic equations. Also, calculating the final speed using F=ma or its relativistic equivalent is complicated because a isn't constant. You have to use calculus.

It's easier if you use conservation of energy. Find the Coulomb potential energy of two protons at whatever distance they exert a force of 14.4 N on each other. Assuming no other forces, and letting them fly apart, when they are far enough apart the potential energy will have (practically) all be converted to kinetic energy, divided equally between the two protons. Calculate the speed from the kinetic energy.

[added: I was surprised to find that the final speed is only about 2% of the speed of light, so if you use conservation of energy, you can safely use the classical formula for kinetic energy. Try it and see!]

 

[PJC01]

I can provide an explanation for this apparent contradiction. Firstly, it is important to understand that Newton's laws and Coulomb's law are both classical laws and do not take into account the principles of special relativity. In the context of nuclear forces, the strong nuclear force overcomes the repulsive force between protons (described by Coulomb's law) due to the exchange of particles called gluons.

The high acceleration calculated using Newton's law is a result of the extremely strong force between protons in the nucleus. However, this does not necessarily mean that the protons will reach speeds exceeding that of light. This is because special relativity states that as an object approaches the speed of light, its mass increases and it requires more and more energy to continue accelerating. This ultimately prevents anything from reaching or exceeding the speed of light.

In the context of nuclear forces, the strong nuclear force is able to overcome the repulsive force between protons without actually accelerating them to speeds that would violate the principles of special relativity. This is due to the unique nature of the strong nuclear force and the exchange of gluons.

In summary, while the force between protons in a nucleus may seem colossal and the acceleration calculated using classical laws may appear to exceed the speed of light, the principles of special relativity still hold true and prevent any actual violations of the speed of light.