A neutron, with kinetic energy K E and moving with velocity...

A neutron, with kinetic energy $K E$ and moving with velocity $v$, collides head on with a nucleus of mass number A perfectly elastically. The fraction of total energy possessed by the neutron transferred to nucleus is (a) $\frac{4 A}{(A-1)^2}$ (b) $\left(\frac{A-1}{A+1}\right)^2$ (c) $\left(\frac{A+1}{2 A}\right)^2$ (d) $\left(\frac{A-1}{2 A}\right)^2$

A neutron, with kinetic energy $K E$ and moving with velocity $v$, collides head on with a nucleus of mass number A perfectly elastically. The fraction of total energy possessed by the neutron transferred to nucleus is
(a) $\frac{4 A}{(A-1)^2}$
(b) $\left(\frac{A-1}{A+1}\right)^2$
(c) $\left(\frac{A+1}{2 A}\right)^2$
(d) $\left(\frac{A-1}{2 A}\right)^2$

Key Concepts

Elastic Collision

In an elastic collision, both kinetic energy and momentum are conserved throughout the interaction. This principle is essential for analyzing collisions between particles, allowing one to derive formulas that describe how velocities and energies are redistributed between the colliding bodies.

Conservation of Momentum

Momentum conservation states that the total momentum of a closed system remains constant if no external forces are acting. This key principle is used to relate the initial and final velocities of the colliding particles, providing a fundamental constraint on the behavior of the system during the collision.

Conservation of Energy

For an elastic collision, the total kinetic energy is conserved in addition to momentum. This makes it possible to equate the sum of the kinetic energies before and after the collision, enabling the determination of how energy is partitioned between the particles involved.

Mass Ratio and Energy Transfer

The fraction of energy transferred during a collision depends critically on the mass ratio of the colliding particles. When a light particle collides elastically with a heavier particle, the formulas derived from momentum and energy conservation show that the energy transfer fraction is proportional to 4mM/(m+M)^2. This expression quantitatively describes how the differing masses influence the efficiency of energy transfer in collisions.

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