The creation and study of new and very massive elementary particles is an important part of contemporary physics. To create a particle of mass $M$ requires an energy $M c^{2} .$ With enough energy, an exotic particle can be created by allowing a fast-moving proton to collide with a similar target particle. Consider a perfectly inelastic collision between two protons: an incident proton with mass $m_{p}$, kinetic energy $K$, and momentum magnitude $p$ joins with an originally stationary target proton to form a single product particle of mass $M .$ Not all the kinetic energy of the incoming proton is available to create the product particle because conservation of momentum requires that the system as a whole still must have some kinetic energy after the collision. Therefore, only a fraction of the energy of the incident particle is available to create a new particle.
(a) Show that the energy available to create a product particle is given by
$$M c^{2}=2 m_{p} c^{2} \sqrt{1+\frac{K}{2 m_{p} c^{2}}}$$
This result shows that when the kinetic energy $K$ of the incident proton is large compared with its rest energy $m_{p} c^{2},$ then $M$ approaches $\left(2 m_{p} K\right)^{1 / 2} / c .$ Therefore, if the energy of the incoming proton is increased by a factor of $9,$ the mass you can create increases only by a factor of $3,$ not by a factor of 9 as would be expected. (b) This problem can be alleviated by using colliding beams as is the case in most modern accelerators. Here the total momentum of a pair of interacting particles can be zero. The center of mass can be at rest after the collision, so, in principle, all the initial kinetic energy can be used for particle creation. Show that
$$M c^{2}=2 m c^{2}\left(1+\frac{K}{m c^{2}}\right)$$
where $K$ is the kinetic energy of each of the two identical colliding particles. Here, if $K>>m c^{2},$ we have $M$ directly proportional to $K$ as we would desire.