2 particles collision - deriving the equation for an energy which can become mass of new particles!

When two particles collide, they can interact in such a way that their kinetic energy is converted into mass, potentially creating new particles. This fascinating process is rooted in Einstein's famous equation, E=mc², which establishes the relationship between energy (E) and mass (m). Let's break down how we derive the equation for the energy required to create new particles during a collision.

The Basics of Particle Collision

In a typical particle collision, two particles approach each other with certain energies. When they collide, they can either bounce off each other or, if the energy is sufficient, combine to form new particles. The key to this transformation lies in the conservation of energy and momentum.

Energy and Mass Relationship

According to Einstein's equation, energy can be converted into mass. This means that if we have enough energy in a collision, we can create mass in the form of new particles. The equation can be rearranged to express mass in terms of energy:

• m = E/c²

Here, c represents the speed of light in a vacuum, approximately 3 x 10⁸ m/s. This relationship shows that even a small amount of energy can produce a significant amount of mass due to the c² factor.

Deriving the Energy Equation

To derive the energy required for new particle creation, we start with the conservation of energy principle. In a collision, the total energy before the collision must equal the total energy after the collision. Let’s denote the initial kinetic energy of the two particles as K₁ and K₂, and the rest mass energies of the particles as E₁ and E₂.

The total energy before the collision can be expressed as:

• E_initial = K₁ + K₂ + E₁ + E₂

After the collision, if new particles are created, the total energy can be expressed as the sum of the rest mass energies of the original particles plus the rest mass energies of the new particles:

• E_final = E₁ + E₂ + E_new

Setting these two equations equal to each other gives us:

• K₁ + K₂ + E₁ + E₂ = E₁ + E₂ + E_new

By simplifying this, we find:

• K₁ + K₂ = E_new

Understanding the Energy Requirement

This equation indicates that the total kinetic energy of the colliding particles must be equal to the energy equivalent of the mass of the new particles created. If we want to create a particle with mass m_new, we can use the earlier relationship:

• E_new = m_new * c²

Combining these insights, we can express the energy required for the creation of new particles as:

• K₁ + K₂ = m_new * c²

Example Scenario

Imagine two protons colliding at high speeds in a particle accelerator. If their combined kinetic energy exceeds the energy equivalent of a new particle (like a meson), then during the collision, this energy can be transformed into the mass of the meson. If the total kinetic energy of the protons is, say, 1 GeV (giga-electronvolt), and the mass of the meson is equivalent to 0.5 GeV/c², the collision has enough energy to create that meson.

This process is fundamental in particle physics and helps us understand the creation of matter in the universe. The energy from collisions not only leads to the formation of new particles but also provides insights into the fundamental forces and interactions that govern the behavior of matter at the smallest scales.