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Chapter 26: Problem 76

A particle decays in flight into two pions, each having a rest energy of 140.0 MeV. The pions travel at right angles to each other with equal speeds of \(0.900 c .\) Find (a) the momentum magnitude of the original particle, (b) its kinetic energy, and (c) its mass in units of \(\mathrm{MeV} / c^{2}\).

Short Answer

Expert verified
Using the given rest energy of 140.0 MeV and the speed of 0.900c for the pions, calculate the following: 1. The momentum magnitude of the original particle before decaying. 2. The kinetic energy of the original particle before decaying. 3. The mass of the original particle in terms of MeV/c². Remember to use the energy-momentum equation and conservation laws to solve this problem.

Step by step solution

01

Find the energy of each pion

We are given that the pions travel at \(0.900c\), where \(c\) is the speed of light, and they each have a rest energy of 140.0 MeV. We can use the relativistic energy-momentum equation to find their total energy: $$ E = \frac{m_0c^2}{\sqrt{1 - \frac{v^2}{c^2}}}, $$ where \(E\) is the total energy, \(m_0\) is the rest mass, \(v\) is the velocity, and \(c\) is the speed of light. Putting the given values in this equation, we get: $$ E = \frac{140.0 \;\text{MeV}}{\sqrt{1 - (0.900)^2}}. $$ Now, we find the energy for each pion, making sure to use the speed of light in terms of MeV\(/c^2\) for consistency with other energy units.
02

Calculate the momentum of each pion

Next, we need to find the momentum of each pion. We can use the energy-momentum equation for this: $$ p = \frac{E}{c}\sqrt{1 - \frac{m_0^2c^4}{E^2}}, $$ where \(p\) is the magnitude of the momentum and \(E\) is the total energy found in step 1. We can plug in the energy values we just found and calculate the momentum for each pion.
03

Find the momentum magnitude of the original particle

Since the pions are moving at right angles, their momenta also make a right triangle. We can use the Pythagorean theorem to find the total momentum magnitude of the original particle before it decayed: $$ p_\text{original} = \sqrt{p_1^2 + p_2^2}. $$ We can plug in the values we found in step 2 for the pion momenta and calculate the momentum magnitude of the original particle.
04

Calculate the total energy of the original particle

The total energy of the original particle is the sum of the energies of the two pions, as the energy is conserved during the decay: $$ E_\text{original} = E_1 + E_2. $$ We can plug in the energy values we found in step 1 for the pion energies and calculate the total energy of the original particle.
05

Find the mass of the original particle

We can now use the energy-momentum equation again to find the mass of the original particle: $$ m_\text{original} = \frac{\sqrt{E_\text{original}^2 - (p_\text{original}c)^2}}{c^2}. $$ We plug in the values we found in step 3 and step 4 for the total energy and momentum of the original particle, respectively, and solve for the mass of the original particle in terms of MeV\(/c^2\).
06

Calculate the kinetic energy of the original particle

Finally, we can calculate the kinetic energy of the original particle by subtracting its rest energy from the total energy found in step 4: $$ K_\text{original} = E_\text{original} - m_\text{original}c^2, $$ where \(K_\text{original}\) is the kinetic energy of the original particle. We can now plug in the values we found in step 4 and step 5 for the original particle's total energy and mass, respectively, and solve for its kinetic energy. After performing these steps, we will have the momentum magnitude, kinetic energy, and mass of the original particle in terms of MeV/c².

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Most popular questions from this chapter

Consider the following decay process: \(\pi^{+} \rightarrow \mu^{+}+v .\) The mass of the pion \(\left(\pi^{+}\right)\) is \(139.6 \mathrm{MeV} / c^{2},\) the mass of the muon \(\left(\mu^{+}\right)\) is \(105.7 \mathrm{MeV} / c^{2},\) and the mass of the neutrino \((v)\) is negligible. If the pion is initially at rest, what is the total kinetic energy of the decay products?
An electron has momentum of magnitude $2.4 \times 10^{-22} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} .$ What is the electron's speed?
Radon decays as follows: $^{222} \mathrm{Rn} \rightarrow^{218} \mathrm{Po}+\alpha .$ The mass of the radon-222 nucleus is 221.97039 u, the mass of the polonium- 218 nucleus is \(217.96289 \mathrm{u},\) and the mass of the alpha particle is 4.00151 u. How much energy is released in the decay? \(\left(1 \mathrm{u}=931.494 \mathrm{MeV} / \mathrm{c}^{2} .\right)\)
Derive the energy-momentum relation $$E^{2}=E_{0}^{2}+(p c)^{2}$$ Start by squaring the definition of total energy \(\left(E=K+E_{0}\right)\) and then use the relativistic expressions for momentum and kinetic energy $[\text{Eqs}. (26-6) \text { and }(26-8)]$.
The mean (average) lifetime of a muon in its rest frame is $2.2 \mu \mathrm{s}\(. A beam of muons is moving through the lab with speed \)0.994 c .$ How far on average does a muon travel through the lab before it decays?
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