An unstable particle at rest spontaneously breaks into two...

An unstable particle at rest spontaneously breaks into two fragments of unequal mass. The mass of the first fragment is $2.50 \times 10^{-28} \mathrm{kg}$ , and that of the other 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

Two-Body Decay Kinematics

Two-body decay kinematics involves analyzing the breakup of a particle into two fragments. In the rest frame of the decaying particle, conservation laws, particularly conservation of momentum, dictate that the magnitudes of the momenta of the fragments are equal while their directions are opposite. This framework is essential in solving problems involving particle decays in high-energy physics.

Conservation of Momentum

In any isolated system, the total momentum is conserved. When a particle decays from rest into two fragments, the momentum of one fragment must be exactly balanced by the momentum of the other in the opposite direction, ensuring that the net momentum remains zero.

Relativistic Momentum

At speeds close to the speed of light, momentum calculations require the relativistic formula p = ?mv, where ? is the Lorentz factor accounting for time dilation and length contraction effects. This concept is crucial when analyzing high-speed particles, as classical momentum expressions are no longer accurate.

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