Question

A particle \(a\) traveling along the positive \(x\) axis of frame \(\mathcal{S}\) with speed 0.5c decays into two identical particles, \(a \rightarrow b+b,\) both of which continue to travel on the \(x\) axis. (a) Given that \(m_{a}=2.5 m_{b},\) find the speed of either \(b\) particle in the rest frame of particle \(a .\) (b) By making the necessary transformation on the result of part (a), find the velocities of the two \(b\) particles in the original frame S.

Short answer

The velocities of the two \(b\) particles in frame S are \(0.846c\) and \(-0.143c\).

Step-by-step solution

Analyze the Decay Process in Rest Frame of Particle a

In the rest frame of particle \(a\), the conservation of momentum must hold. Since particle \(a\) is at rest, the two \(b\) particles must have equal and opposite velocities to conserve momentum.Let the velocity of one \(b\) particle be \(v'_b\) in the rest frame of \(a\). Since the particles are identical and the total momentum is zero, \(v'_b = -v'_b\). We need to solve for \(v'_b\) using conservation of energy.

Apply Conservation of Energy in Rest Frame of Particle a

In the rest frame of particle \(a\), energy is conserved:\[ m_a c^2 = 2 \times \gamma_b m_b c^2 \]where \(\gamma_b\) is the Lorentz factor for particle \(b\):\[ \gamma_b = \frac{1}{\sqrt{1 - \left(\frac{v'_b}{c}\right)^2}} \]Given that \(m_a = 2.5 m_b\), we can substitute and solve:\[ 2.5 m_b c^2 = 2 \times \gamma_b m_b c^2 \]\[ \frac{2.5}{2} = \gamma_b \]\[ \gamma_b = 1.25 \]

Calculate the Velocity of b Particles in Rest Frame of a

Use \(\gamma_b = 1.25\) to find \(v'_b\):\[ \gamma_b = \frac{1}{\sqrt{1 - \left(\frac{v'_b}{c}\right)^2}} \]Solve for \(v'_b\):\[ 1.25 = \frac{1}{\sqrt{1 - \left(\frac{v'_b}{c}\right)^2}} \]\[ \sqrt{1 - \left(\frac{v'_b}{c}\right)^2} = \frac{1}{1.25} \]\[ 1 - \left(\frac{v'_b}{c}\right)^2 = \left(\frac{4}{5}\right)^2 \]\[ \left(\frac{v'_b}{c}\right)^2 = 1 - \frac{16}{25} \]\[ \left(\frac{v'_b}{c}\right)^2 = \frac{9}{25} \]\[ v'_b = 0.6c \]

Perform Lorentz Transformation to Find Velocities in Frame S

In frame \(S\), particle \(a\) has velocity \(u = 0.5c\) and the velocities of the \(b\) particles need to be calculated. Using velocity addition formula:\[ v_{b1} = \frac{v'_b + u}{1 + \frac{v'_b u}{c^2}} \]\[ v_{b2} = \frac{-v'_b + u}{1 - \frac{v'_b u}{c^2}} \]Substitute \(v'_b = 0.6c\):\[ v_{b1} = \frac{0.6c + 0.5c}{1 + \frac{0.6c \times 0.5c}{c^2}} = \frac{1.1c}{1.3} = 0.846c \]\[ v_{b2} = \frac{-0.6c + 0.5c}{1 - \frac{0.6c \times 0.5c}{c^2}} = \frac{-0.1c}{0.7} = -0.143c \]

Conclusion

The velocity of the \(b\) particle in the rest frame of \(a\) is \(0.6c\). In frame \(S\), the velocities of the \(b\) particles are \(0.846c\) and \(-0.143c\).

Key concepts

Lorentz Transformation

Understanding the Lorentz Transformation is essential in the realm of relativistic physics, where objects move at speeds close to the speed of light. This transformation allows us to convert the coordinates of an event from one inertial frame to another.

When transitioning between frames, the Lorentz transformation helps calculate how measurements of space and time vary. It accounts for the effects of time dilation and length contraction. The transformations are:

• For time: \( t' = \gamma (t - \frac{vx}{c^2}) \)
• For position: \( x' = \gamma (x - vt) \)

Where \( \gamma \) is the Lorentz factor, \( v \) is the relative speed between the frames, \( t \) is time in the stationary frame, and \( x \) is the position.

In practical problems like our exercise, using the Lorentz transformation simplifies finding the velocities in different frames, ensuring consistent results across varying perspectives.

Conservation of Momentum

In physics, momentum conservation is a fundamental principle that states the total momentum of a closed system remains constant over time, if it is not influenced by external forces. In the context of our exercise, momentum conservation is crucial when dealing with particle decays.

When particle \( a \) decays into two identical particles \( b \), the system's momentum before and after the decay must remain unchanged. In the rest frame of particle \( a \), which initially has zero momentum, the sum of the momenta of the two particles \( b \) must also be zero.

This leads to the condition:
\[ m_b v'_b + m_b (-v'_b) = 0 \]
This condition emphasizes that both particles travel at equal speeds but opposite directions, ensuring no net change in momentum.

Conservation of Energy

Within any closed physical system, the energy conservation principle maintains that the total energy remains constant over time. This principle is vital in analyzing high-speed particle interactions in relativistic mechanics.

In the given exercise, during the decay of particle \( a \) into two particles \( b \), the initial energy of particle \( a \) is entirely converted into the energies of the two \( b \) particles. This can be expressed as:

• Initial energy: \( E_a = m_a c^2 \)
• Final energy: \( 2 \times \gamma_b m_b c^2 \)

Equating these gives: \[ m_a c^2 = 2 \gamma_b m_b c^2 \] Substituting the given mass relationship, \( m_a = 2.5 m_b \), lets us solve for \( \gamma_b \): \[ \gamma_b = \frac{1.25}{1} \] This equation ensures that we respect the energy conservation rule, allowing us to find missing variables like velocity in the decay process.

Velocity Addition Formula

Velocity addition in Special Relativity differs from classical physics, due to the involvement of the speed of light. The relativistic velocity addition formula is an essential tool when dealing with moving reference frames.

In our scenario, to find particle \( b \)'s velocity in the original stationary frame \( S \), where particle \( a \) was initially traveling, we use the formula:

• Velocity of particle \( b \) moving in direction of particle \( a \):\[ v_{b1} = \frac{v'_b + u}{1 + \frac{v'_b u}{c^2}} \]
• Velocity of particle \( b \) moving opposite the direction of particle \( a \):\[ v_{b2} = \frac{-v'_b + u}{1 - \frac{v'_b u}{c^2}} \]

In this formula, \( v'_b \) is the velocity in the rest frame of particle \( a \), \( u \) is the velocity of the frame, and \( c \) is the speed of light.

By substituting known values, we ensure all velocities are calculated accurately, adhering to relativistic principles. The formula confirms that as velocities approach the speed of light, they never surpass it, honoring Einstein's principle of relativity.