Chapter 15: Problem 74
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
Step by step solution
Analyze the Decay Process in Rest Frame of Particle a
Apply Conservation of Energy in Rest Frame of Particle a
Calculate the Velocity of b Particles in Rest Frame of a
Perform Lorentz Transformation to Find Velocities in Frame S
Conclusion
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Lorentz Transformation
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: \( x' = \gamma (x - vt) \)
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
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
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: \( 2 \times \gamma_b m_b c^2 \)
Velocity Addition Formula
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 \( a \):\[ v_{b2} = \frac{-v'_b + u}{1 - \frac{v'_b u}{c^2}} \]
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.