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The reference frame K^' moves in the positive direction of the x axis of the frame K with a relative velocity V^'. Suppose that at the moment when the origins of coordinates O and O^' coincide, the clock readings at these points are equal to zero in both frames. Find the displacement velocity x of the point (in the frame K) at which the readings of the clocks of both reference frames will be permanently identical. Demonstrate that overset(.)x lt V. |
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Answer» SOLUTION :SUPPOSE `x(t)` is the locus of points in the frame K at which the readings of the clocks of both reference system are permanently identical, then by Lorentz transformation `t^'=(1)/(sqrt(1-V^2//c^2))(t-(VX(t))/(c^2))=t` So differentiating `x(t)=c^2/V(1-sqrt(1-V^2/c^2))=c/beta(1-sqrt(1-beta^2)), beta=V/c` Let `beta=tan h theta, 0 le theta lt oo`, Then `x(t)=(c)/(tanhtheta)(1-sqrt(1-tan h^2theta))=c(coshtheta)/(sinhtheta)(1-(1)/(coshtheta))` `=c(coshtheta-1)/(sinhtheta)=csqrt((coshtheta-1)/(coshtheta+1))=ctanhtheta/2lev` (`tan htheta` is a monotonically increasing FUNCTION of `theta`) |
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