91Ó°ÊÓ

Momentum is an essential concept in physics, especially when analyzing collisions. In simple terms, momentum is the product of an object's mass and velocity. For a system of particles, the total momentum before an event must equal the total momentum after the event, if no external forces are acting on it. This principle is known as the Conservation of Momentum.
To think about this more concretely, consider two colliding particles: a neutron and a nucleus. Before the collision, the neutron is moving, and the nucleus is at rest. Therefore, only the neutron has momentum. Let’s denote the mass of the neutron as \(m_n\) and its initial velocity as \(u\). The momentum of the neutron before the collision is \(m_n \times u\).
After the collision, both the neutron and the nucleus move with new velocities, let’s call them \(v_n\) and \(v_{nucleus}\) for the neutron and nucleus respectively. The Conservation of Momentum equation for this system would be:
\[m_n \times u = m_n \times v_n + Am_n \times v_{nucleus}\]
This equation tells us that the mass of each object and its respective velocity determine how momentum is shared between them after a collision. Here, \(A\) represents the mass number of the nucleus, thus \(Am_n\) is its mass.
Using this principle helps us understand how velocity changes and how kinetic energy is redistributed between the objects involved after a collision.

In physics, the Conservation of Energy states that energy cannot be created or destroyed, only converted from one form to another. In elastic collisions, like the one between the neutron and nucleus, kinetic energy is conserved. This means that the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
Initially, only the neutron has kinetic energy since the nucleus is at rest. Using the formula for kinetic energy \(K = \frac{1}{2}mv^2\), we can write the initial kinetic energy for the neutron as \(\frac{1}{2}m_n u^2\). After the collision, the system must still have the same amount of total kinetic energy, shared between the neutron and nucleus:
\[\frac{1}{2}m_n u^2 = \frac{1}{2}m_n v_n^2 + \frac{1}{2}Am_n v_{nucleus}^2\]
This equation represents the conservation of kinetic energy, showing us how it is distributed between the neutron and nucleus after their interaction. As we solved in the problem, knowing the velocities post-collision allows us to find the retained kinetic energy of the neutron by substituting back the known expressions for velocity.

Kinetic energy is the energy an object possesses due to its motion. The formula for kinetic energy is \(K = \frac{1}{2} m v^2\), where \(m\) is the mass and \(v\) is the velocity of the object. In the context of elastic collisions, analyzing changes in kinetic energy helps us understand how energy is transferred between colliding bodies.
In our neutron and nucleus example, initially, the neutron possesses kinetic energy, which is partially transferred to the nucleus upon collision. To calculate the fraction of kinetic energy retained by the neutron, we first find its velocity after the collision:\[v_n = \frac{u(1 - A)}{1 + A}\]
We then compute the new kinetic energy \(K'\) using this velocity:
\[K' = \frac{1}{2}m_n \left(\frac{u(1 - A)}{1 + A}\right)^2\]
By formulating the ratio of \(K'\) over the initial kinetic energy \(K\), \(\left(\frac{1-A}{1+A}\right)^2\), we find this fraction of kinetic energy the neutron retains. This exercise demonstrates how velocity outcomes directly impact kinetic energy distribution in a system.