PX155 - I5 - relativistic kinetic energy

we have:

E_{K} = \int F . d x = \int \frac{d p}{d t} \frac{d x}{d t} . d t = \int v \frac{d p}{d t} . d t

integrating by parts:

E_{K} = v p - \int p \frac{d v}{d t} . d t = v p - \int p . d v

integrating from $0$ to $v$ : $$E_{K}= [vp]{0}^{v} - \int^{v}p.dv$$

E_{K} = \frac{m_{0} v^{2}}{(1 - \frac{v^{2}}{c^{2}})^{\frac{1}{2}}} - \int_{0}^{v} \frac{m_{0} v}{(1 - \frac{v^{2}}{c^{2}})^{\frac{1}{2}}} . d v

here,

\frac{d}{d v} (1 - \frac{v^{2}}{c^{2}})^{\frac{1}{2}} = - 2 \frac{v}{c^{2}} \frac{1}{2} (1 - \frac{v^{2}}{c^{2}})^{- \frac{1}{2}} = - \frac{v}{c^{2}} \frac{1}{(1 - \frac{v^{2}}{c^{2}})^{\frac{1}{2}}}

we have:

E_{K} = \frac{m_{0} v^{2}}{(1 - \frac{v^{2}}{c^{2}})^{\frac{1}{2 - -}}} + m_{0} c^{2} {[(1 - \frac{v^{2}}{c^{2}})^{\frac{1}{2}}]}_{0}^{v}

E_{K} = \frac{m_{0} v^{2}}{(1 - \frac{v^{2}}{c^{2}})^{\frac{1}{2}}} + m_{0} c^{2} (1 - \frac{v^{2}}{c^{2}})^{\frac{1}{2}} - m_{0} c^{2}

E_{K} = γ m_{0} c^{2} - m_{0} c^{2}

∴ E_{K} = (γ - 1) m_{0} c^{2}

equivalence of mass and energy

here, to particles of rest mass, $m_{0}$ , were collided to form a new particle with $M_{0} = 2 γ m_{0}$
increase in rest mass was:

Δ m_{0} = 2 (γ - 1) m_{0} = \frac{Δ E_{K}}{c^{2}}

- before collision:

E_{K} = 2 (γ - 1) m_{0} c^{2}

- after collision:

E_{K} = 0

Δ E_{K} = 2 (γ - 1) m_{0} c^{2} = Δ m_{0} c^{2}

if something already has a rest mass of $m_{0}$ , then an amount of energy must have gone into creating it. the rest mass energy:

E_{0} = m_{0} c^{2}

total energy (kinetic + rest mass): $$E = E_{0} + E_{K} = m_{0}c^{2} + (\gamma-1)m_{0}c^{2} = \gamma m_{0}c^{2}$$

∴ E = m c^{2}

eg: the sun's luminosity, $L = 3.8 \times 10^{26} W$ . at what rate does it lose mass?
$E = m c^{2}$
- so:$$\dot m = \frac{\dot E}{c^{2}} = - \frac{L}{c^{2}} = -4.3 \times10^{9} kg,s^{-1}$$
eg: calculate speed of an electron which has $E_{K} = 30 k e V$ ( $E_{0}$ for electron is $511 k e V$ )
$E_{K} = (γ - 1) m_{0} c^{2} = (γ - 1) E_{0}$ $γ - 1 = \frac{E_{K}}{E_{0}} = 1 + \frac{30}{511}$ $v = c \sqrt{1 - \frac{1}{γ^{2}}} = 0.33 c$