Mechanics Berkeley Physics

Charles Kittel, Walter D. Knight, Malvin A. Ruderman, A. Carl Helmholz, Burton J. Moyer

Chapter 12

Relativistic Dynamics: Momentum and Energy - all with Video Answers

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Chapter Questions

Problem 1

Relativistic momentum. What is the momentum of a proton having kinetic energy $1 \mathrm{BeV} ?$ (If $E$ is measured in $\mathrm{BeV}$, we may measure $p$ in $\mathrm{BeV} / c$.

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Problem 2

Relativistic momentum. What is the momentum of an electron having kinetic energy $1 \mathrm{BeV}$ ? $\quad$

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Problem 3

Photon momentum. What is the momentum of a photon of energy 1 BeV?

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Problem 4

Energy and momentum of fast proton. Given a proton whose $\beta=0.995$ measured in the laboratory; what are the corresponding relativistic total energy and momentum? What is the kinetic energy?

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Problem 5

Energetic cosmic-ray particles. It is known that cosmic-ray particles have energies up to $10^{19} \mathrm{eV}$, and perhaps higher.
(a) What is the apparent mass of such a particle (approximately)?
(b) What is the momentum (approximately)?

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Problem 6

Transformation of energy and momentum. A proton has $\beta=0.999$ in the laboratory. Find the energy and momentum as observed in a frame traveling in the same direction, with $\beta^{\prime}=0.990$ with respect to the laboratory.

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Problem 7

Energy of fast electron. An electron has $\beta=0.99 .$ What is its kinetic energy?

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Problem 8

Recoil in \gamma-ray emission. What is the recoil momentum in the laboratory of an $\mathrm{Fe}^{57}$ nucleus recoiling due to the emission of a 14 -keV photon? Is the momentum of the nucleus relativistic? $\quad$

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Problem 9

Consider a $\gamma$ -ray of energy $E_{\gamma}$ directed toward a proton at rest in the laboratory.
(a) In the laboratory frame what is the momentum of the $\gamma$ -ray?
(b) Show that the velocity $V$ of the center of mass in the laboratory frame is given by
$$
\frac{V}{c}=\frac{E_{\gamma}}{E_{\gamma}+M_{p} c^{2}}
$$
(c) What is the \gamma-ray energy in the center-of-mass frame? Also the proton energy in the center-of-mass frame?

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Problem 10

Neutron decay. Use values given in Chap. 12 to calculate the amount of energy released as a neutron decays into a proton and an electron.

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Problem 11

Lorentz invariance in two-particle system. Let the total momentum and energy for a two-particle system be $\mathrm{p}=\mathrm{p}_{1}+\mathrm{p}_{2}$ and $E=E_{1}+E_{2}$, respectively. Show explicitly that the Lorentz transformations on $\mathrm{p}$ and $E$ are consistent with the invariance of the quantity $E^{2}-p^{2} c^{2}$

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Problem 12

Transformation to rest frame from center-of-mass reference frame. Two protons travel in opposite directions from a common point with velocities $\beta=0.5$.
(a) What are the energy and momentum of one proton relative to the common point?
(b) Use the Lorentz transformation to find the energy and momentum of one proton in the rest frame of the other. (In problems of this sort, it is usually convenient to express the energy as a multiple of some rest mass energy.)

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Problem 13

Radiation of mass by radio transmitter. What is the mass equivalent of the energy from an antenna radiating $1000 \mathrm{~W}$ of radio energy for $24 \mathrm{~h}$ ? ( $1 \mathrm{~W} \equiv 10^{7}$ ergs $/ \mathrm{s}$.)

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Problem 14

Solar energy. The solar constant is the flux of solar energy per square centimeter per second at the distance of the earth from the sun. By measurement it is found that the value of the constant is $1.4 \times 10^{6} \mathrm{ergs} / \mathrm{s}-\mathrm{cm}^{2}$
(a) Show that the total energy generation of the sun is $\approx 4 \times 10^{33} \mathrm{ergs} / \mathrm{s}$
(b) Show that the average rate of energy generation per gram of matter on the sun is $\approx 2$ ergs $/ \mathrm{g}-\mathrm{s} \approx 6 \times 10^{7}$ ergs/g-yr.
(c) Show that the energy equivalent of $1 \mathrm{~g}$ of hydrogen burned to produce $\mathrm{He}^{4}$ is $\approx 6 \times 10^{18}$ ergs.
(d) Show that if the mass of the sun were one-third hydrogen and the nuclear burning process continued without change, then the sun could continue to radiate at its present rate for $3 \times 10^{10} \mathrm{yr}$.

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Problem 15

Radiation propulsion. One possible means of propulsion in space is a large reflecting metallic sheet connected to a small vehicle. Make reasonable estimates of the accelerations which might result for some typical vehicle at a distance of $1 \mathrm{AU}$ from the sun.

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Problem 16

Momentum of laser pulse. A large laser can produce a pulse of light having an energy of $2000 \mathrm{~J} .\left(1 \mathrm{~J} \equiv 10^{7}\right.$ ergs.)
(a) Show that the momentum of the pulse is of the order of $1 \mathrm{~g}-\mathrm{cm} / \mathrm{s}$
(b) Discuss how you might detect this momentum. The duration of the pulse might be $1 \mathrm{~ms}\left(10^{-3} \mathrm{~s}\right)$.

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