Question

More significant than the kinematic features of the special theory of relativity are the dynamical processes that it describes that Newtonian dynamics does not. Suppose a hypothetical particle with rest mass \(1.000 \mathrm{GeV} / c^{2}\) and \(\mathrm{ki}-\) netic energy \(1.000 \mathrm{GeV}\) collides with an identical particle at rest. Amazingly, the two particles fuse to form a single new particle. Total energy and momentum are both conserved in the collision. a) Find the momentum and speed of the first particle. b) Find the rest mass and speed of the new particle.

Short answer

Answer: The momentum and speed of the first particle are \(p_1 = \dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\) and \(v_1 = c\,\dfrac{\sqrt{1- \dfrac{(1.000\,\mathrm{GeV}/c^2)^2}{(\sqrt{3}\,\mathrm{GeV}/c)^2}}}{\sqrt{3}}\), while the rest mass and speed of the new particle are \(m_f = \dfrac{\sqrt{(3.000\,\mathrm{GeV})^2 - \left(\dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\right)^2c^2}}{c^2}\) and \(v_f = c\,\dfrac{\sqrt{1 - \dfrac{m_f^2c^2}{\left(\dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\right)^2}}}{m_f}\).

Step-by-step solution

Find the energy of the first particle

We know the rest mass energy (\(E_0\)) and the kinetic energy (KE) of the first particle. The total energy of the first particle can be found by adding these two energies. Total energy (E) = Rest mass energy (E_0) + Kinetic energy (KE) = \(1.000\,\mathrm{GeV}+1.000\,\mathrm{GeV}=2.000\,\mathrm{GeV}\)

Calculate the momentum of the first particle

Now, we will find the momentum of the first particle using energy-momentum relationship: \(E^2=p^2c^2+E_0^2\) Solving for momentum (p), we get \(p=\dfrac{\sqrt{E^2 - E_0^2}}{c}\) Plug in the values: \(p=\dfrac{\sqrt{(2.000\,\mathrm{GeV})^2 - (1.000\,\mathrm{GeV})^2}}{c}=\dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\)

Calculate the speed of the first particle

Now, we will find the speed of the first particle using the momentum and rest mass: \(p=\dfrac{mv}{\sqrt{1 - \dfrac{v^2}{c^2}}}\) Solving for the speed (v), we get \(v=mc\dfrac{\sqrt{1 - \dfrac{v^2}{c^2}}}{p}\) Replacing \(m\) and \(p\) with their values: \(v=c\,\dfrac{\sqrt{1- \dfrac{(1.000\,\mathrm{GeV}/c^2)^2}{(\sqrt{3}\,\mathrm{GeV}/c)^2}}}{\sqrt{3}}\)

Find the total energy and momentum of two initial particles

Since particle 2 is at rest, its kinetic energy is 0. The total energy and momentum of two particles are the sum of their individual energies and momenta. Total energy = \(E_1 + E_2 = (2.000 + 1.000)\,\mathrm{GeV} = 3.000\,\mathrm{GeV}\) Total momentum = \(p_1 + p_2 = \dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\)

Calculate the rest mass of the new particle

Let's denote the rest mass of the final particle as \(m_f\). We know the total energy and momentum of the two initial particles are conserved. To find \(m_f\), we will use the energy-momentum relation: \(E^2 = p^2c^2 + m_f^2c^4\) Plug in the values of total energy and total momentum: \((3.000\,\mathrm{GeV})^2 = \left(\dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\right)^2c^2 + m_f^2c^4\) Solving for \(m_f\): \(m_f = \dfrac{\sqrt{(3.000\,\mathrm{GeV})^2 - \left(\dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\right)^2c^2}}{c^2}\)

Calculate the speed of the new particle

Since the total momentum of the system is conserved, the momentum of the resulting particle, denoted as \(p_f\), is equal to the total momentum before the collision. Knowing \(p_f\) and \(m_f\), we can find the speed of the new particle using the momentum formula: \(p_f=\dfrac{m_fv_f}{\sqrt{1 - \dfrac{v_f^2}{c^2}}}\) Again, solving for speed (\(v_f\)), we get \(v_f=c\,\dfrac{\sqrt{1 - \dfrac{m_f^2c^2}{\left(\dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\right)^2}}}{m_f}\) By obtaining these results, we found: a) The momentum and speed of the first particle: \(p_1 = \dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\) and \(v_1 = c\,\dfrac{\sqrt{1- \dfrac{(1.000\,\mathrm{GeV}/c^2)^2}{(\sqrt{3}\,\mathrm{GeV}/c)^2}}}{\sqrt{3}}\) b) The rest mass and speed of the new particle: \(m_f = \dfrac{\sqrt{(3.000\,\mathrm{GeV})^2 - \left(\dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\right)^2c^2}}{c^2}\) and \(v_f = c\,\dfrac{\sqrt{1 - \dfrac{m_f^2c^2}{\left(\dfrac{\sqrt{3}\,\mathrm{GeV}}{c}\right)^2}}}{m_f}\)

Key concepts

Energy-Momentum Relation

The energy-momentum relation is an essential concept in special relativity. It ties together the energy and momentum of a particle, taking into account both its rest mass and kinetic energy. The relationship is expressed as:

\(E^2 = p^2c^2 + E_0^2\)

Where:

• \(E\) is the total energy of the particle.
• \(p\) is the momentum.
• \(c\) is the speed of light.
• \(E_0\) is the rest mass energy, calculated as \(m_0c^2\).

This equation shows that a particle's momentum and energy are intertwined, especially when speaking of particles moving at high velocities. The equation allows us to calculate one parameter if the others are known, which is particularly useful in analyzing particles involved in collisions.

Particle Collision

In particle physics, a collision refers to an event where two or more particles interact, often resulting in a change of properties or the creation of new particles. During a collision like the one described in the exercise, significant transformations can occur.

For instance, two particles merging to form a single new particle is a phenomenon made possible by absorbing kinetic energy into mass energy, or altering the kinetic and potential energy of the involved particles. Such events are vital in the study of particle physics as they illustrate fundamental principles, such as the conversion of energy into mass, as predicted by Einstein's famous equation \(E = mc^2\).

These collisions help us understand the forces and processes that dictate particle behavior on a cosmic scale, providing insights into both the micro and macro aspects of the universe.

Conservation of Momentum

The conservation of momentum is a fundamental principle in physics indicating that the total momentum of a closed system remains constant, provided no external forces act upon it. The scenario from the exercise highlights this idea, demonstrating that the momentum before and after a particle collision is conserved.

In mathematical terms, this principle can be expressed as:
\( p_{initial} = p_{final} \)

Where:

• \(p_{initial}\) is the total momentum before the collision.
• \(p_{final}\) is the total momentum after the collision.

This concept implies that the momentum exchanged in a system results in no net loss or gain, reinforcing the understanding of balanced interactions within that system. During particle collisions, this conservation law helps us track and predict the events occurring as particles interact.

Rest Mass Energy

Rest mass energy is a term in relativity that relates the mass of a particle to its energy, irrespective of its motion. The iconic equation \(E_0 = m_0c^2\), formulated by Einstein, expresses this intrinsic energy stored as 'mass.'

Understanding rest mass energy is crucial when analyzing situations where energy converts into mass and vice versa, such as in high-speed particle collisions.

This energy component describes the energy contribution simply due to the object's mass, without having to consider motion or kinetic variables. In particle physics, calculating rest mass energy is foundational to understanding how particles interact and transform, as it represents an invariant quantity that does not change regardless of the reference frame.

Kinetic Energy

Kinetic energy (KE) in the realm of relativity takes on more complexity than in classical physics. It describes the additional energy a particle has due to its motion. Kinetic energy can be expressed as the difference between total energy and rest mass energy:

\( KE = E - E_0 \)

In high-speed scenarios, kinetic energy contributes significantly to a particle's overall energy budget.

• As a particle's velocity approaches the speed of light, its kinetic energy increases dramatically.
• This aspect is critical during particle collisions, where kinetic energy can manifest as new particles.

Understanding kinetic energy in the context of special relativity is vital when evaluating interactions occurring at those immense speeds, such as the ones found in particle accelerators.