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Unformatted text preview: 62. (a) Eq. 2′ of Table 37-2, becomes ∆t′ = γ(∆t − β∆x/c) = γ[1.00 µs − β(240 m)/(2.998 × 102 m/µs )]
= γ (1.00 − 0.800 β ) µ s
where the Lorentz factor is itself a function of β (see Eq. 37-8).
(b) A plot of ∆t′ is shown for the range 0 < β < 0.01 : (c) A plot of ∆t′ is shown for the range 0.1 < β < 1 : (d) The minimum for the ∆t′ curve can be found from by taking the derivative and
simplifying and then setting equal to zero:
= γ3(β∆t – ∆x/c) = 0 .
Thus, the value of β for which the curve is minimum is β = ∆x/c∆t = 240/299.8, or
β = 0.801 .
(e) Substituting the value of β from part (d) into the part (a) expression yields the
minimum value ∆t′ = 0.599 µs. (f) Yes. We note that ∆x/∆t = 2.4 ×108 m/s < c. A signal can indeed travel from event A
to event B without exceeding c, so causal influences can originate at A and thus affect
what happens at B. Such events are often described as being “time-like separated” – and
we see in this problem that it is (always) possible in such a situation for us to find a frame
of reference (here with β ≈ 0.801) where the two events will seem to be at the same
location (though at different times). ...
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This note was uploaded on 06/03/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.
- Spring '08