Ranking Lorentz factors (three particles)

Problem

Three particles travel at speeds:

• A: $v = 0{,}30\,c$;
• B: $v = 0{,}80\,c$;
• C: $v = 0{,}99\,c$.

Rank the Lorentz factors from smallest to largest.

Solution

Formula. $\gamma = 1/\sqrt{1-\beta^2}$, an increasing function of $\beta$: the faster the particle, the greater $\gamma$. So it suffices to rank the speeds.

Numerical values.

$\gamma_A = \frac{1}{\sqrt{1-0{,}30^2}} = \frac{1}{\sqrt{0{,}91}} \approx 1{,}05,$

$\gamma_B = \frac{1}{\sqrt{1-0{,}80^2}} = \frac{1}{\sqrt{0{,}36}} \approx 1{,}67,$

$\gamma_C = \frac{1}{\sqrt{1-0{,}99^2}} = \frac{1}{\sqrt{0{,}0199}} \approx 7{,}09.$

Ranking.

$\ev{\gamma_C > \gamma_B > \gamma_A}$

Note the non-linear growth: going from $0{,}80\,c$ to $0{,}99\,c$ the factor quadruples, because the curve $\gamma(v)$ diverges as it approaches $c$.

Links

Topics: Relatività ristretta Concepts: Dilatazione dei tempi Skills: Ragionamento di ranking