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!]