Question

Find the speed an alpha particle requires to come within 4.0 ✕ 10−14 m of an iron nucleus. Find the energy of the alpha particle in MeV.

Answer 1

We can solve this problem using conservation of energy. Initially alpha particle is moving at speed (kinetic energy ) towards the iron nucleus, and since it is very far away from the nucleus the potential energy of the proton is zero. Total initial energy is

At the point of closest approach, the proton is momentarily at rest that is its kinetic energy is zero, since it is very close to the iron nucleus it's potential energy is not negligible

The total energy at this point is

Using , , , and

We get

By conservation of energy

In eV,

Now we shall check if the alpha particle is relativistic.

The rest mass energy of alpha particle is

Comparing kinetic energy with rest mass energy

Since kinetic energy is much smaller than rest mass energy, we can safely use classical kinetic energy formula

Using and

We get

Note that if kinetic was comparable to the rest mass energy we would have to use the relativistic kinetic energy formula.