Fusion may occur if you have deuterium (an isotope of hydrogen with 1 proton and 1 neutron atoms colliding at speeds near the speed of light from opposite directions in a cyclotron (a circular particle accelerator).

Here are the equations:
\[
\text {Kinetic Energy} E_{k1}=\frac {1}{2}mv^2, E_{k2}=\frac{1}{2}mv^2\\
\text {Since the atoms are travelling in opposite directions, total energy} E_T=E_{k1}+E_{k2}\\
=\frac{1}{2}mv^2+\frac{1}{2}mv^2=mv^2.\\
\text{If } v \rightarrow c, \text{then total energy } E=mc^2,\\
\text{which implies that matter is being converted into energy.}\\
\text{Sorry that everything is centered. I couldn't figure out how to end or add a line.}
\]
 

Edited November 14, 20178 yr by IsaacAsimov
I'm not that familiar with LaTeX math.

 

 Your Latex didn't display

That being pointed out,  your equations are incorrect.

while

KE =  mv²/2

in Newtonian physics, this is actually only a close approximation to the correct answer for cases where v is small compared to the speed of light.

The correct formula is

KE = mc² (sqrt(1-v²/c²)-1)

E=mc² on the other hand, is energy equivalent of the rest mass, and has nothing to do with how fast the object is moving.( It is the total energy you could theoretically get from converting matter to energy)  The total energy would be the sum of the "rest" energy equivalence and KE or

E_t = mc²/sqrt(1-v²/c²)

With hydrogen fusion,  the 2 Deuterium nuclei combine to form a Helium nucleus, which has a slightly less rest mass than the 2 Deuterium nuclei had combined (~ 1% less).  This "lost" rest mass is what is converted to energy, both as radiation and increased kinetic energy of the product.  The KE of the collision is only involved in so far as it allows the Deuterium nuclei to overcome their mutual repulsion and get close enough to fuse.

So in fusion you start with a certain rest mass and KE, and end up with a bit less rest mass and more KE.  Exactly how much rest mass is converted to energy depends on the nature of the fusing nuclei.  (There is more than one hydrogen fusion process)

 

Edited November 14, 20178 yr by Janus