Question

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 c² . 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 mp , 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.(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 thatM c² =2 m c² ((1+1/km /c²))where K is the kinetic energy of each of the two identical colliding particles. Here, if K>>m c² , we have M directly proportional to K as we would desire.

Answer 1

Consider a collision between two protons that is perfectly inelastic: an incident proton with mass m(p), kinetic energy K, and momentum magnitude p combines with a target proton that was initially stationary to form a single product particle with mass M.

Now,

In this case, the problem is alleviated by using colliding beams.

A pair of interacting particles' combined momentum may be zero.

After the collision, the center of mass is at rest.

Therefore, all the initial kinetic energy can be used to create particles.

Let K be the kinetic energy of each of the two identical colliding particles.

Therefore, the energy released by the two colliding beams are:

E₁ = K + mc² and E₂ = K + mc²

In the final state,

E(f) = Mc² where M is the new mass of the single product after the collision.

By conservation of energy,

E₁ + E₂ =E(f)

K + mc² + K + mc² = Mc²

2K + 2mc² = Mc²

Taking 2mc² common, we get

Mc² = 2mc²[ 1 + (K/mc²)]

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