PX155 - I7 - massless particles

if $m_{0} = 0$ , then, $E^{2} - p^{2} c^{2} = 0$ , and $E = p c$ (or strictly, $E = | p | c$ )
the only sure-fire example of a massless particle is the photon
photon have energy and so momentum, $\frac{E}{c}$
eg: a spacecraft equipped with a photon drive annihilates half its rest mass (stored as matter-antimatter mix) directing resulting photons in a narrow beam. how much speed is gained starting from rest?
- conserving energy: $$E_{S} + E_{p} = m_{0}c^{2}; ...(1)$$ $$E_{S}^{2} - p_{S}^{2}c^{2} = \left( \frac{m_{0}}{2} \right)^{2}c^{4} ; ...(2)$$

E_{p} = p_{p} c . . . (3)

- conserving momentum:

| p_{S} | = | p_{p} | . . . (4)

- $(4)$ in $(3)$ :

E_{p} = p_{S} c

- in $(2)$ :

E_{S}^{2} - E_{p}^{2} = (E_{S} + E_{p}) (E_{S} - E_{p}) = {(\frac{m_{0}}{2})}^{2} c^{4}

(E_{S} - E_{p}) m_{0} c^{2} = \frac{m_{0}^{2}}{4} c^{4}

(E_{S} - E_{p}) = \frac{1}{4} m_{0} c^{2}

- add $(1)$ and divide by $2$ :

E_{S} = \frac{5}{4} (\frac{m_{0}}{2}) c^{2} = γ (\frac{m_{0}}{2}) c^{2} = γ E_{0}

γ = \frac{5}{4} ⟹ v = 0.6 c