CM1_HW9

11.47

Let $A$ , $B$ , and $C$ be frames of reference with coordinates $(c t, x)$ , $(c \tilde{t}, \tilde{x})$ , and $(c \overline{t}, \overline{x})$ . Let the velocity of $B$ be $v_{1}$ with respect to $A$ , and the velocity of $C$ be $v_{2}$ with respect to $B$ . If we let $β_{v_{1}} = v_{1} / c$ , $β_{v_{2}} = \frac{v _{2}}{c}$ , $γ_{v_{1}} = 1/ 1 - β_{v_{1}}^{2}$ , and $γ_{v_{2}} = 1/ 1 - β_{v_{2}}^{2}$ , we can write

[c \tilde{t} \tilde{x}] = γ_{v_{1}} [1 - β_{v_{1}} - β_{v_{1}} 1] [c t x] [c \overline{t} \overline{x}] = γ_{v_{2}} [1 - β_{v_{2}} - β_{v_{2}} 1] [c \tilde{t} \tilde{x}]

⟹ [c \overline{t} \overline{x}] = γ_{v_{1}} γ_{v_{2}} [1 - β_{v_{2}} - β_{v_{2}} 1] [1 - β_{v_{1}} - β_{v_{1}} 1] [c t x] .

On simplifying, we get

[c \overline{t} \overline{x}] = \frac{1}{1 - β _{v_{1}}^{2} 1 - β _{v_{2}}^{2}} [1 + β_{v_{1}} β_{v_{2}} - β_{v_{2}} - β_{v_{1}} - β_{v_{1}} - β_{v_{2}} 1 + β_{v_{1}} β_{v_{2}}] [c t x] = \frac{1}{1 - β ^{2}} [1 - β - β 1] [c t x]

where

β = \frac{β _{v_{1}} + β _{v_{2}}}{1 + β _{v_{1}} β _{v_{2}}} .

On substituting the values for $β_{v_{1}}$ and $β_{v_{2}}$ and equating to $v / c$ , we get

\frac{v}{c} = \frac{\frac{v _{1}}{c} + \frac{v _{2}}{c}}{1 + \frac{v _{1} v _{2}}{c ^{2}}} ⟹ v = \frac{v _{1} + v _{2}}{1 + v _{1} v _{2} / c ^{2}}

NoNotes

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CM1_HW9

11.47

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