Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / Section B / B9

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- Solution B9

Solution

For the stated standard configuration, the Lorentz transformation is

t^{'} = γ (t - vx / c^{2}), x^{'} = γ (x - v t), y^{'} = y, z^{'} = z, γ = (1 - v^{2} / c^{2})^{- 1/2} .

Differentiate along the moving object's worldline and divide

d x^{'}

by

d t^{'}

to obtain the velocity-addition formula:

u_{x}^{'} = \frac{u _{x} - v}{1 - u _{x} v / c ^{2}} .

For a receding source, two emissions separated by proper time

1/ f^{'}

occur

γ / f^{'}

apart in S. The second signal travels an additional distance

v γ / f^{'}

, so its reception interval is

γ (1 + v / c) / f^{'}

. Hence the longitudinal relativistic Doppler effect is

f = \frac{f ^{'}}{γ ( 1 + v / c )} = f^{'} \frac{1 - v / c}{1 + v / c} .

The inverse velocity transformation for the forward throw gives

u = \frac{c /2 + c /7}{1 + 1/14} = \frac{3}{5} c .

The ball approaches the stationary receiver, so its forward light is blueshifted. The received wavelength is

400 (1 - 3/5) / (1 + 3/5) = 200 nm

, in the ultraviolet. The stated monochromatic emission is outside the visible band. Other reflected or emitted frequencies have not been specified. ^[ref]

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