REL-08

Relativistic momentum

p = γ m v, E² = (p c)² + (m c²)².

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Special relativityEinstein 1905

Governing equation

p=\gamma mv,\quad E=\gamma mc^2

$v/c \in \left[0, 0.999\right]$

where

m: Rest mass (u)
v/c: Speed in c (—)
p: Momentum (MeV/c)
E: Total energy (MeV)

The figure, part by part

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①

Governing law — red

p=\gamma mv,\quad E=\gamma mc^2

$v/c \in \left[0, 0.999\right]$

$v/c = 0$ $p = 931.5$

Along the curve

②

Domain of the variable — blue

v/c_{\min} = 0,\quad v/c_{\max} = 0.999

$v/c \in \left[0, 0.999\right]$

$v/c = 0$ $v/c = 0$

Along the curve

③

Response curve — green

p=\gamma mv,\quad E=\gamma mc^2

$v/c \in \left[0, 0.999\right]$

$v/c = 0$ $p = 931.5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}p}{\mathrm{d}v/c}

$v/c \in \left[0, 0.999\right]$

$v/c = 0.006$ $p^{\prime} = 5.853$

Along the curve

Lecture brief

Historical brief

Einstein’s 1905 Lorentz kinematics and 1915 field equation, then Schwarzschild (1916) and Hawking temperature, recast time, mass and gravity. The lab computes dilation, E=mc² and horizon scales. This sheet (REL-08 — Relativistic momentum) is the form associated with Einstein 1905. Working symbols: $m$ , $v/c$ $\rightarrow$ $p$ , $E$ . Four-momentum conservation requires p = γ m v. The mass-shell relation E² − p²c² = m²c⁴ is the invariant.

Purpose

Purpose: compute

p

E

from

m

v/c

in Relativity via

p=\gamma mv,\quad E=\gamma mc^2

p = γ m v, E² = (p c)² + (m c²)². Use it when a real relativity question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

m = 1.000\,\mathrm{u}

v/c = 0.500\,\mathrm{—}

, the governing relation

p=\gamma mv,\quad E=\gamma mc^2

yields

p = 537.798\,\mathrm{MeV/c}

E = 1075.597\,\mathrm{MeV}

. Massive particle. Enter v/c and m in u. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Rest mass m1 uSpeed in c v/c0.5 —

Outputs

Momentum p537.798 MeV/c
Total energy E1075.597 MeV

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Narration of this film

Massive particle. Enter v/c and m in u.

Four-momentum conservation requires p = γ m v. The mass-shell relation E² − p²c² = m²c⁴ is the invariant.

Reading speed

Watch on YouTube