The creation and study of new elementary particles is an important part of contemporary physics. Especially interesting is the discovery of a very massive particle. 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 particle of ordinary matter, such as a proton, to collide with a similar target particle. Let us 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 $M$ You might think that the creation of a new product particle, nine times more massive than in a previous experiment, would require just nine times more energy for the incident proton. Unfortunately not all of the kinetic energy of the incoming proton is available to create the product particle, since conservation of momentum requires that after the collision the system as a whole still must have some kinetic energy. Only a fraction of the energy of the incident particle is thus available to create a new particle. You will determine how the energy available for particle creation depends on the energy of the moving proton. Show that the energy available to create a product particle is given by
$$
M c^{2}=2 m_{p} c^{2} \sqrt{1+\frac{K}{2 m_{p} c^{2}}}
$$
From this result, when the kinetic energy $K$ of the incident proton is large compared to its rest energy $m_{p} c^{2},$ we see that $M$ approaches $\left(2 m_{p} K\right)^{1 / 2} / c .$ Thus if the energy of the incoming proton is increased by a factor of nine, the mass you can create increases only by a factor of three. This disappointing result is the main reason that most modern accelerators, such as those at CERN (in Europe), at Fermi lab (near Chicago), at SLAC (at Stanford), and at DESY (in Germany), use colliding beams. 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 of the initial kinetic energy can be used for particle creation, according to
$$
M c^{2}=2 m c^{2}+K=2 m c^{2}\left(1+\frac{K}{2 m c^{2}}\right)
$$
where $K$ is the total kinetic energy of two identical colliding particles. Here if $K>m c^{2}$, we have $M$ directly proportional to $K,$ as we would desire. These machines are difficult to build and to operate, but they open new vistas in physics.