A certain galaxy has a Doppler shift given by ƒ₀ - ƒ = 0.1015 ƒ₀. Estimate how fast it is moving away from us.

An atomic clock is taken to the North Pole, while another stays at the Equator. How far will they be out of synchronization after 1.5 years has elapsed? [Hint: Use the binomial expansion, Appendix A–2.]

(III) (a) In reference frame S, a particle has momentum $\overrightarrow{p}=p_{x}i$ along the positive x axis. Show that in frame S’, which moves with speed v as in Fig. 36–12, the momentum has components

$p_x^{\prime }=\frac{px-vE/c^2}{\sqrt{1-v^2/c^2}}$

$p_y^{\prime }=py$

$p_z^{\prime }=pz$

$E^{\prime }=\frac{E-p_{x}v}{\sqrt{1-v^2/c^2}}.$

(These transformation equations hold, actually, for any direction of $\overrightarrow{p}$, as long as the motion of S' is along the x axis.) (b) Show that p_x, p_y, p_z, E/c transform according to the Lorentz transformation in the same way as x, y, z, ct.

Make a graph of the kinetic energy versus momentum for (a) a particle of nonzero mass, and (b) a particle with zero mass.

A spaceship moving toward Earth at 0.65c transmits radio signals at 95.0 MHz. At what frequency should Earth receivers be tuned?

An electron (m = 9.11 x 10⁻³¹ kg) is accelerated from rest to speed v by a conservative force. In this process, its potential energy decreases by 7.20 x 10⁻¹⁴ J . Determine the electron’s speed, v.

Show that the kinetic energy K of a particle of mass m is related to its momentum p by the equation $p=\frac{\sqrt{K^2+2Km{c}^2}}{c}$.