Problem — Ranking Lorentz factors (four spaceships)

Problem

Ranking. Four spaceships move relative to the laboratory with speeds: (A) $v = 0{,}3\,c$; (B) $v = 0{,}6\,c$; (C) $v = 0{,}8\,c$; (D) $v = 0{,}95\,c$. Rank the Lorentz factors $\gamma = 1/\sqrt{1 - v^2/c^2}$ from smallest to largest.

Solution

Qualitative observation. $\gamma$ is a monotonically increasing function of $v$: the faster the spaceship, the greater $\gamma$. So the order of the $\gamma$‘s follows the order of the speeds.

Calculating the values.

$\gamma_A = \frac{1}{\sqrt{1 - 0{,}09}} = \frac{1}{\sqrt{0{,}91}} \approx 1{,}05$

$\gamma_B = \frac{1}{\sqrt{1 - 0{,}36}} = \frac{1}{\sqrt{0{,}64}} = 1{,}25$

$\gamma_C = \frac{1}{\sqrt{1 - 0{,}64}} = \frac{1}{\sqrt{0{,}36}} \approx 1{,}67$

$\gamma_D = \frac{1}{\sqrt{1 - 0{,}9025}} = \frac{1}{\sqrt{0{,}0975}} \approx 3{,}20$

Ranking.

$\ev{\gamma(A) < \gamma(B) < \gamma(C) < \gamma(D)}$

Note how $\gamma$ grows slowly at low speeds (from $0{,}3\,c$ to $0{,}6\,c$ it only goes from $1{,}05$ to $1{,}25$) but then explodes as it approaches $c$ (at $0{,}95\,c$ it is already $3{,}2$).

Links

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