The use of the conservation laws for momentum and energy is also very important in particle collisions. This is a very powerful rule because it can allow us to determine the results of a collision without knowing the details of the collision. The law of conservation of momentum states that the total momentum is conserved in the collision of two objects such as billiard balls. The assumption of conservation of momentum and the conservation of kinetic energy makes possible the calculation of the final velocities in two-body collisions. At this point, we have to distinguish between two types of collisions:
- Elastic collisions
- Inelastic collisions
Elastic Collisions
A perfectly elastic collision is defined as one in which there is no net conversion of kinetic energy into other forms (such as heat or noise). For the brief moment during which the two objects are in contact, some (or all) of the energy is stored momentarily in the form of elastic potential energy. But if we compare the total kinetic energy just before the collision with the total kinetic energy just after the collision, they are found to be the same. We say that the total kinetic energy is conserved.
- Large-scale interactions between satellites and planets are perfectly elastic, like the slingshot type gravitational interactions (also known as a planetary swing-by or a gravity-assist maneuver).
- Collisions between very hard spheres may be nearly elastic, so it is useful to calculate the limiting case of an elastic collision.
- Collisions in ideal gases approach perfectly elastic collisions, as do scattering interactions of sub-atomic particles deflect by the electromagnetic force.
- Rutherford scattering is the elastic scattering of charged particles also by the electromagnetic force.
- A neutron-nucleus scattering reaction may also be elastic, but in this case, the neutron is deflected by a strong nuclear force.
The relative speed of the two objects after the collision has the same magnitude (but opposite direction) as before the collision, no matter what the masses are.
It is known the fission neutrons are of importance in any chain-reacting system. All neutrons produced by fission are born as fast neutrons with high kinetic energy. Before such neutrons can efficiently cause additional fissions, they must be slowed down by collisions with nuclei in the reactor’s moderator. The probability of the fission U-235 becomes very large at the thermal energies of slow neutrons. This fact implies an increase in the multiplication factor of the reactor (i.e., lower fuel enrichment is needed to sustain a chain reaction).
The neutrons released during fission with an average energy of 2 MeV in a reactor on average undergo many collisions (elastic or inelastic) before they are absorbed. During the scattering reaction, a fraction of the neutron’s kinetic energy is transferred to the nucleus. It is possible to derive the following equation for the mass of target or moderator nucleus (M), the energy of incident neutron (Ei), and the energy of scattered neutron (Es) using the laws of conservation of momentum and energy and the analogy of collisions of billiard balls for elastic scattering.
where A is the atomic mass number, in case of the hydrogen (A = 1) as the target nucleus, the incident neutron can be completely stopped. But this works when the direction of the neutron is completely reversed (i.e., scattered at 180°). In reality, the direction of scattering ranges from 0 to 180 °, and the energy transferred also ranges from 0% to maximum. Therefore, the average energy of the scattered neutron is taken as the average of energies with scattering angles 0 and 180°.
Moreover, it is useful to work with logarithmic quantities. Therefore one defines the logarithmic energy decrement per collision (ξ) as a key material constant describing energy transfers during a neutron slowing down. ξ is not dependent on energy, only on A and is defined as follows:
For heavy target nuclei, ξ may be approximated by the following formula:
From these equations, it is easy to determine the number of collisions required to slow down a neutron from, for example from 2 MeV to 1 eV.
Example: Determine the number of collisions required for thermalization for the 2 MeV neutrons in the carbon.
ξCARBON = 0.158
N(2MeV → 1eV) = ln 2⋅106/ξ =14.5/0.158 = 92
For a mixture of isotopes:
What are the velocities of the neutron and carbon nucleus after the collision?
Solution:
This is an elastic head-on collision of two objects with unequal masses. We have to use the conservation laws of momentum and kinetic energy and apply them to our system of two particles.
We can solve this system of equations or use the equation derived in the previous section. This equation stated that the relative speed of the two objects after the collision has the same magnitude (but opposite direction) as before the collision, no matter what the masses are.
The minus sign for v’ tells us that the neutron scatters back of the carbon nucleus because the carbon nucleus is significantly heavier. On the other hand, its speed is less than its initial speed. This process is known as neutron moderation, and it significantly depends on the mass of moderator nuclei.
Inelastic Collisions
An inelastic collision is when part of the kinetic energy is changed to some other form of energy in the collision. Any macroscopic collision between objects will convert some kinetic energy into internal energy and other forms of energy, so no large-scale impacts are perfectly elastic. For example, in collisions of common bodies, such as two cars, some energy is always transferred from kinetic energy to other forms of energy, such as thermal energy or energy of sound. The inelastic collision of two bodies always involves a loss in the kinetic energy of the system. The greatest loss occurs if the bodies stick together, in which case the collision is called a completely inelastic collision. Thus, the system’s kinetic energy is not conserved, while the total energy is conserved as required by the general principle of conservation of energy. Momentum is conserved in inelastic collisions, but one cannot track the kinetic energy through the collision since some of it is converted to other forms of energy.
In nuclear physics, an inelastic collision is when the incoming particle causes the nucleus to strike to become excited or break up. Deep inelastic scattering is a method of probing the structure of subatomic particles in much the same way as Rutherford probed the inside of the atom (see Rutherford scattering).
In nuclear reactors, inelastic collisions are of importance in the neutron moderation process. An inelastic scattering plays an important role in slowing down neutrons, especially at high energies and by heavy nuclei. Inelastic scattering occurs above threshold energy. This threshold energy is higher than the energy of the first excited state of the target nucleus (due to the laws of conservation), and it is given by the following formula:
Et = ((A+1)/A)* ε1
where Et is known as the inelastic threshold energy and ε1 is the energy of the first excited state. Therefore especially scattering data of 238U, which is a major component of nuclear fuel in commercial power reactors, are one of the most important data in the neutron transport calculations in the reactor core.
A ballistic pendulum is a device for measuring the velocity of a projectile, such as a bullet. The ballistic pendulum is a kind of “transformer,” exchanging the high speed of a light object (the bullet) for the low speed of a massive object (the block). When a bullet is fired into the block, its momentum is transferred to the block. The bullet’s momentum can be determined from the amplitude of the pendulum swing.
When the bullet is embedding itself in the block, it occurs so quickly that the block does not move appreciably. The supporting strings remain nearly vertical, so negligible external horizontal force acts on the bullet–block system, and the horizontal component of momentum is conserved. However, mechanical energy is not conserved during this stage because a nonconservative force does work (the force of friction between bullet and block).
In the second stage, the bullet and block move together. The only forces acting on this system are gravity (a conservative force) and string tensions (which do not work). Thus, as the block swings, mechanical energy is conserved. Momentum is not conserved during this stage because there is a net external force (the forces of gravity and string tension don’t cancel when the strings are inclined).
Equations governing the ballistic pendulum
In the first stage, momentum is conserved, and therefore:
where v is the initial velocity of the projectile of mass mP. v’ is the velocity of the block and embedded projectile (both of mass mP + mB) just after the collision, before they have moved significantly.
In the second stage, mechanical energy is conserved. We choose y = 0 when the pendulum hangs vertically and then y = h when the block and embedded projectile system reach maximum height. The system swings up and comes to rest for an instant at a height y, where its kinetic energy is zero, and the potential energy is (mP + mB)gh. Thus we write the law of conservation of energy:
which is the initial velocity of the projectile and our final result.
When we use some realistic numbers:
- mP = 5 g
- mB = 2 kg
- h = 3 cm
- v = ?
then we have:
