Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 4 / 11A / c

Table of contents
- Solution c

Solution

t_{0}

, the accelerated proton has momentum

p = qE t_{0}

and, by the relativistic energy-momentum relation, energy

E_{p} = m^{2} c^{4} + q^{2} E^{2} t_{0}^{2} c^{2} .

The incident Cosmic microwave background photon adds energy

E_{γ}

. Since the final state contains only the Delta-plus baryon, conservation of relativistic energy gives its laboratory-frame Lorentz factor

γ_{Δ} = \frac{E _{p} + E _{γ}}{m _{Δ} c ^{2}} .

Its rest-frame decay time

t_{Δ}

is a proper time, so time dilation gives the laboratory decay time

t_{lab} = \frac{t _{Δ}}{m _{Δ} c ^{2}} (m^{2} c^{4} + q^{2} E^{2} t_{0}^{2} c^{2} + E_{γ}) .

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