Question

A particle of mass \(m_{a}\) decays at rest into two identical particles each of mass \(m_{b} .\) Use conservation of momentum and energy to find the speed of the outgoing particles.

Short answer

The speed of the outgoing particles is \( v = c \sqrt{1 - \left(\frac{2m_b}{m_a}\right)^2} \).

Step-by-step solution

Understanding Conservation Laws

To solve this problem, we need to apply the conservation of energy and momentum. Since the particle starts at rest, its initial momentum is zero. The particle has an initial rest energy given by its mass-energy equivalence, which is the mass of the initial particle multiplied by the speed of light squared.

Initial Energy Calculation

The initial energy of the decaying particle is given by the rest energy formula: \[ E_a = m_a c^2 \] where \( m_a \) is the mass of the initial decaying particle, and \( c \) is the speed of light.

Final Energy and Momentum

After decay, the energy is distributed between the two identical particles. Each of these particles has a total energy consisting of rest energy and kinetic energy. If the velocity of each of the particles is \( v \), then the energy for each particle is given by:\[ E_b = \gamma m_b c^2 \]where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) is the Lorentz factor.

Momentum Conservation

Since the initial particle is at rest, the total initial momentum of the system is zero. After decay, the momentum of the two particles must also add up to zero. If they move in opposite directions with speed \( v \), their momenta are equal in magnitude but opposite in direction: \[ p_b = \gamma m_b v \]Thus, the total momentum is: \[ 2 \gamma m_b v = 0 \] This highlights the importance of the concept that they must be opposite.

Equation Relating Energy Conservation

Since the total energy after decay must equal the initial energy, we equate the initial energy with the sum of the energies of the two particles:\[ m_a c^2 = 2 \gamma m_b c^2 \]Solving for \( \gamma \), we have:\[ \gamma = \frac{m_a}{2m_b} \]

Solve for Velocity

Now, using \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \), we can solve for \( v \):\[ \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{m_a}{2m_b} \]Squaring both sides:\[ 1 - \frac{v^2}{c^2} = \left(\frac{2m_b}{m_a}\right)^2 \]Solving for \( v \):\[ \frac{v^2}{c^2} = 1 - \left(\frac{2m_b}{m_a}\right)^2 \]\[ v = c \sqrt{1 - \left(\frac{2m_b}{m_a}\right)^2} \]

Finalize the Calculation

The above equations provide the speed \( v \) of the two particles post-decay, considering they move in opposite directions to maintain zero total momentum. The key concept is recognizing that the decaying process maintains the conservation of both energy and momentum.

Key concepts

Conservation of Momentum

In the realm of physics, particularly in understanding particle decay, the conservation of momentum stands as a foundational principle. When an object or system is in a certain state, the total momentum before and after any event, like a decay, must remain constant, given there are no external forces acting on it. This is crucial in particle decay scenarios, such as our example where a particle of mass \(m_a\) decays into two particles each of mass \(m_b\). Since the particle starts at rest, its initial momentum is zero.

Upon decay, these two identical particles shoot out in opposite directions. This is essential for maintaining the total momentum of zero, as required by the conservation principle. Each particle contributes a momentum that is equal in magnitude but opposite in direction to the momentum contributed by the other particle. In mathematical terms, if the velocity of each particle is \(v\), the total momentum equation post-decay is represented as:

• \(2 \gamma m_b v = 0\)

This equation ensures that the magnitudes of the momenta cancel each other out, perfectly illustrating momentum conservation in a closed system.

Mass-Energy Equivalence

The concept of mass-energy equivalence, encapsulated by Einstein's iconic equation \(E = mc^2\), is paramount in understanding particle decay. It implies that mass and energy are two forms of the same thing and can be converted into each other. In the context of a particle decay, the rest energy of a particle is defined by its mass multiplied by the speed of light squared.

For the decaying particle of mass \(m_a\), its initial energy is strictly in the form of rest energy since it is at rest, calculated as:

• \(E_a = m_a c^2\)

After decay, this energy is transformed into the combined rest and kinetic energy of the two resulting particles, each contributing an energy \(E_b = \gamma m_b c^2\). This transformation upholds the total energy conservation, as the entire energy of the system pre-decay must equal the total energy post-decay:

• \(m_a c^2 = 2 \gamma m_b c^2\)

This equation shows that mass has been redistributed into the new particles, linking closely with the fundamental idea that energy cannot be lost, only changed in form.

Lorentz Factor

In the realm of relativistic physics, the Lorentz factor \(\gamma\) emerges as a critical component in particle decay processes. It accounts for the effects of relativistic speeds—those approaching the speed of light. As particles accelerate to significant fractions of the speed of light, their behavior departs from what classical mechanics predicts.The Lorentz factor is defined by the equation:

• \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\)

Here, \(v\) represents the velocity of the particle, and \(c\) is the speed of light. This factor scales the mass-energy of a moving particle, explaining phenomena such as time dilation and length contraction.

In our decay scenario, the Lorentz factor modifies the energy and momentum of the newly formed particles. By solving \(\gamma = \frac{m_a}{2m_b}\), we derive how close the velocities of those particles approach the limit set by the speed of light. This highlights how the innate properties of mass and the finite speed limit of the universe intertwine to influence outcomes in particle physics.

Relativistic Kinetics

As particles reach velocities close to the speed of light, classical measures such as kinetic energy and momentum transition to relativistic kinetics. This field adjusts classical equations to accommodate relativism, ensuring both conservation laws hold true for extremely fast-moving particles.In classical physics, kinetic energy is calculated as \(KE = \frac{1}{2}mv^2\). However, in the relativistic regime, we adjust this to consider the Lorentz factor as well:

• \(E = \gamma mc^2\)

This equation shows that not only does a moving particle have more energy due to its velocity, but its effective mass also changes. Because of this, the calculation of the particle's velocity needs to factor in the Lorentz transformations.In our decay problem, solving the equation:

• \(v = c \, \sqrt{1 - \left(\frac{2m_b}{m_a}\right)^2}\)

provides us with the velocity of the particles post-decay. This result emerges directly from the synthesis of conservation laws and the relativistic reinterpretation of classical kinetic measures, ensuring all physics principles align even under extreme conditions.