When an particle collides elastically with a nucleus, the nucleus recoils. A 5.00-MeV particle has a head-on elastic collision with a gold nucleus, initially at rest. We have to find the kinetic energy (a) of the recoiling nucleus and (b) of the rebounding particle. The mass of the particle may be taken to be 4.00 u and that of the gold nucleus 197 u.

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We will use principles of conservation of energy and momentum for answering this problem. Let M be the mass of gold nucleus and m be the mass of particle. Let E be the kinetic energy of the incident particle, and let P be the momentum of the recoiling gold nucleus and let p be the momentum of the rebounding particle.

As the collision is elastic we will require that the sum of the kinetic energies of the recoiling gold nucleus and the rebounding particle be equal to the kinetic energy of the incident particle.

Momentum of the incident particle, .

From the conservation of momentum we get the following equation:

And from the conservation of energy, we get the following equation:

From the above two equations, after algebraic simplification, we get

There are two solutions for . The first one implies that the gold nucleus continues to remain at rest after collision with the particle, which contradicts the statement of the problem. We therefore select the second solution

We substitute the data:

We find

The kinetic energy of the recoiling gold nucleus will therefore be

From conservation of energy, we note that the kinetic energy of the rebounding particle will be