Chapter 15: Problem 75
A particle of unknown mass \(M\) decays into two particles of known masses \(m_{a}=0.5 \mathrm{GeV} / c^{2}\) and \(m_{b}=1.0 \mathrm{GeV} / c^{2},\) whose momenta are measured to be \(\mathbf{p}_{a}=2.0 \mathrm{GeV} / \mathrm{c}\) along the \(x_{2}\) axis and \(\mathbf{p}_{b}=1.5 \mathrm{GeV} / c\) along the \(x_{1}\) axis. \(\left(1 \mathrm{GeV}=10^{9} \mathrm{eV} .\right)\) Find the unknown mass \(M\) and its speed.
Short Answer
Step by step solution
Calculate Energy of Particle a
Calculate Energy of Particle b
Calculate Total Energy and Momentum of Initial Particle
Find Unknown Mass M of Initial Particle
Calculate Speed of Initial Particle
Conclusion: Unknown Mass and Speed
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Energy-Momentum Relation
- \( E = \sqrt{(m c^2)^2 + (pc)^2} \)
For our case with particle \( a \) and \( b \), using this relation helps us find their respective energies knowing their masses and momenta.
Knowing the energy of the final state particles is key as it helps find the initial state quantities such as total energy and momentum, later used to find the invariant mass.
Invariant Mass
The invariant mass can be determined using the total energy and momentum of the system through:
- \( M c^2 = \sqrt{E^2 - (pc)^2} \)
This formula tells us that even though the energy and momentum can vary based on the observer's frame of reference, the invariant mass is constant.
In our particle decay example, knowing the decay products' energies and momenta helps compute the original particle's mass.
Particle Decay
In the decay process considered in the problem, a particle of unknown mass decays into two particles with known masses and measurable momenta.
Important points to remember about particle decay:
- Conservation of Energy: Total energy before decay equals total energy after.
- Conservation of Momentum: Total momentum before decay equals total momentum after.
Relativistic Speed
The speed of a particle can be found using the relation:
- \( E = \gamma M c^2 \)
- \( p = \gamma M v \)
In our calculation, the unknown particle's speed approximates to 0.660c, indicating significant relativistic effects due to its high velocity.
Understanding relativistic speed ensures accurate analysis of a particle's behavior at speeds nearing that of light, crucial in high-energy physics experiments.