# The rest mass of an alpha particle is 6.7 \times 10^{-27} kg. If an alpha particle has a kinetic...

## Question:

The rest mass of an alpha particle is {eq}6.7 \times 10^{-27} {/eq} kg. If an alpha particle has a kinetic energy of 6 MeV, what is its speed?

## Relativistic Energy:

The total energy of a fast-moving object is the product of the rest energy of the object and Lorentz factor. The total energy is the sum of the rest energy and the kinetic energy in relativity. The rest energy of the object is the product of the rest mass of the object and the speed of light squared.

According to the relativity by Einstein, the kinetic energy K is expressed as

{eq}K=E-E_o\\ \rm Here:\\ \,\,\,\, \, \bullet \,E(=mc^2/\sqrt{1-v^2/c^2}) \text{: total energy}\\ \,\,\,\, \, \bullet \,E_o(=mc^2) \text{: rest energy}\\ \,\,\,\, \, \bullet \,m(=6.7\times 10^{-27}\, kg) \text{: mass of the alpha particle}\\ \,\,\,\, \, \bullet \,v\text{: speed of the alpha particle} \,\,\,\, \, \bullet \, c(=3\times10^8\, m/s)\text{: speed of light}\\ {/eq}

The kinetic energy is given as 6 MeV. Please note, 1 eV is equivalent to {eq}1.6\times 10^{-19}\, J {/eq}.

Solving for v, we have

{eq}\begin{align} v&=\dfrac{\sqrt{K+2Kmc^2}}{K+mc^2}\, c\\\\ &=\dfrac{\sqrt{6\times 10^6\times 1.6\times 10^{-19}\, +2\times 6\times 10^6\times 1.6\times 10^{-19}\times 6.7\times 10^{-27}\times (3\times10^8)^2}}{6\times 10^6\times 1.6\times 10^{-19}+6.7\times 10^{-27}\times (3\times10^8)^2}\, \times 3\times10^8\\\\ &=\boxed{1.69\times 10^7\, m/s} \end{align} {/eq}