An unstable particle at rest breaks into two fragments of...

An unstable particle at rest breaks into two fragments of unequal mass. The mass of the lighter fragment is $2.50 \times 10^{-28} \mathrm{~kg}$, and that of the heavier fragment is $1.67 \times 10^{-27} \mathrm{~kg}$. If the lighter fragment has a speed of $0.893 c$ after the breakup, what is the speed of the heavier fragment?

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Key Concepts

Conservation of Momentum

In any isolated system, the total momentum is conserved. For a particle at rest that decays into two fragments, this means that the momentum of one fragment must exactly counterbalance the momentum of the other, ensuring that the vector sum of their momenta remains zero.

Relativistic Momentum

When dealing with particles moving at speeds comparable to the speed of light, the classical definition of momentum (p = mv) is inadequate. The correct definition in such cases is the relativistic momentum, given by p = ?mv, where ? is the Lorentz factor. This formula accounts for the increase in momentum due to relativistic effects.

Lorentz Factor

The Lorentz factor, defined as ? = 1/?(1 - (v/c)²), is a key component in special relativity. It quantifies how time, length, and mass measurements change for objects moving at significant fractions of the speed of light, and it is essential for accurately calculating relativistic momentum and energy.

Relativistic Decay Kinematics

This concept involves applying both conservation laws and relativistic mechanics to processes where unstable particles decay into fragments that may travel at relativistic speeds. It requires the use of relativistic formulas for momentum and energy to analyze and determine the properties of the resulting particles accurately.

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