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Chapter 20: Problem 31

Find the speed parameter \(\beta\) and the Lorentz factor \(\gamma\) for a particle whose kinetic energy is \(10 \mathrm{MeV}\) if the particle is \((a)\) an electron, \((b)\) a proton, and \((c)\) an alpha particle.

Short Answer

Expert verified
The Lorentz factor and speed parameter for the three particles are calculated individually by using their respective rest masses. They are calculated using the formulas for kinetic energy and speed parameter in special relativity.

Step by step solution

01

Determine the Rest Mass

Determine the rest mass for each particle. The rest mass of an electron (me) is approximately \(0.511 \mathrm{MeV}/c^2\), of a proton (mp) is about \(938.27 \mathrm{MeV}/c^2\), and of an alpha particle (mα) is about \(3727.4 \mathrm{MeV}/c^2\).
02

Calculate the Lorentz Factor

Calculate the Lorentz factor \(\gamma\) for each particle using the kinetic energy formula. Inversely, the formula \(\gamma = \frac{KE}{mc^2} + 1\) can be used. KE is given as \(10 \mathrm{MeV}\). Plug the values into the formula to get the value of \(\gamma\).
03

Calculate the Speed Parameter

Calculate the speed parameter \(\beta\) using the formula \(\beta^2 = 1 - \frac{1}{\gamma^2}\). After calculating the Lorentz factor in the previous step, substitute \(\gamma\) in the formula to get \(\beta\).

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Most popular questions from this chapter

An unstable high-energy particle enters a detector and leaves a track \(1.05 \mathrm{~mm}\) long before it decays. Its speed relative to the detector was \(0.992 c .\) What is its proper lifetime? That is, how long would it have lasted before decay had it been at rest with respect to the detector?

Observer \(S\) assigns to an event the coordinates \(x=100 \mathrm{~km}\), \(t=200 \mu \mathrm{s}\). Find the coordinates of this event in frame \(S^{\prime}\), which moves in the direction of increasing \(x\) with speed \(0.950 c\). Assume that \(x=x^{\prime}\) at \(t=t^{\prime}=0\).

An alpha particle with kinetic energy \(7.70 \mathrm{MeV}\) strikes a \({ }^{14} \mathrm{~N}\) nucleus at rest. An \({ }^{17} \mathrm{O}\) nucleus and a proton are produced, the proton emitted at \(90^{\circ}\) to the direction of the incident alpha particle and carrying kinetic energy \(4.44 \mathrm{MeV}\). The rest energies of the various particles are: alpha particle, \(3730.4 \mathrm{MeV}\); \({ }^{14} \mathrm{~N}, 13,051 \mathrm{MeV} ;\) proton, \(939.29 \mathrm{MeV} ;{ }^{17} \mathrm{O}, 15,843 \mathrm{MeV}\) (a) Find the kinetic energy of the \({ }^{17} \mathrm{O}\) nucleus. (b) At what angle with respect to the direction of the incident alpha particle does the \({ }^{17} \mathrm{O}\) nucleus move?

A spaceship whose rest length is \(358 \mathrm{~m}\) has a speed of \(0.728 c\) with respect to a certain reference frame. A micrometeorite, with a speed of \(0.817 c\) in this frame, passes the spaceship on an antiparallel track. How long does it take this micrometeorite to pass the spaceship?

Calculate the speed of a particle \((a)\) whose kinetic energy is equal to twice its rest energy and \((b)\) whose total energy is equal to twice its rest energy.
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