Okay, let's try to figure out this problem. So, the question is about finding the kinetic energy of the product particles when a Pi-plus meson (Pi(+) ) decays into a muon (mu(+)) and a neutrino (nu). The Pi(+) is stationary, and we know their rest masses: Pi(+) is 139.6 MeV/c², and the muon is 105.7 MeV/c². The choices are given, and I need to pick the correct pair of kinetic energies for the muon and the neutrino.

Hmm, right. So, first, I remember that when a particle at rest decays into two particles, the energy and momentum must be conserved. Since the initial momentum is zero (because Pi(+) is stationary), the total momentum of the decay products must also be zero. That means the muon and the neutrino must have equal and opposite momenta. So, their momenta are equal in magnitude but opposite in direction.

Now, energy conservation comes into play. The total energy before decay is just the rest energy of the Pi(+) meson, which is 139.6 MeV. After decay, the total energy is the sum of the energies of the muon and the neutrino. The energy of each particle is the sum of their rest mass energy and their kinetic energy. However, wait—the neutrino is a massless particle, right? Because neutrinos have very tiny masses, but in many cases, especially in particle physics problems like this, they are treated as massless. If that's the case, the neutrino's energy is entirely kinetic, so E_nu = p_nu * c, but since we're working in units where c=1 (MeV and MeV/c²), the energy of the neutrino would just be equal to its momentum times c, which simplifies to E_nu = p_nu. But wait, actually, for massless particles, the energy is |p| * c, but since we're using natural units where c=1, it's just |p|. So, E_nu = p_nu.

Similarly, the muon has a rest mass, so its energy is E_mu = sqrt(p_mu² + m_mu²), where m_mu is the rest mass of the muon. But since the neutrino's momentum is equal in magnitude to the muon's (because total momentum must be zero), p_nu = p_mu. Let's denote this common momentum as p. So, E_mu = sqrt(p² + m_mu²), and E_nu = p.

The total energy after decay is E_mu + E_nu = sqrt(p² + m_mu²) + p. This must equal the rest energy of the Pi(+) meson, which is 139.6 MeV. So, we can set up the equation:

sqrt(p² + (105.7)^2) + p = 139.6

Our goal is to solve for p, and then find the kinetic energies, which would be the total energy minus the rest mass energy. For the muon, the kinetic energy K_mu is E_mu - m_mu = sqrt(p² + (105.7)^2) - 105.7. For the neutrino, since it's massless, the kinetic energy K_nu is E_nu - 0 = p.

But wait, the question asks for the kinetic energy of the product particles. So, the two kinetic energies are K_mu and K_nu. The options are pairs of numbers, so we need to find which pair (K_mu, K_nu) matches the given choices.

Let me write down the equation again:

sqrt(p² + (105.7)^2) + p = 139.6

Let me solve for p. Let's let me denote m_mu = 105.7 MeV/c². So, the equation becomes:

sqrt(p² + m_mu²) + p = M_pi, where M_pi = 139.6 MeV/c².

Let me rearrange the equation:

sqrt(p² + m_mu²) = M_pi - p

Now, square both sides:

p² + m_mu² = (M_pi - p)^2 = M_pi² - 2 M_pi p + p²

Subtract p² from both sides:

m_mu² = M_pi² - 2 M_pi p

Then, solve for p:

2 M_pi p = M_pi² - m_mu²

p = (M_pi² - m_mu²) / (2 M_pi)

Wait, that's interesting. Let me plug in the numbers:

M_pi = 139.6 MeV

m_mu = 105.7 MeV

So, p = (139.6² - 105.7²) / (2 * 139.6)

First compute 139.6 squared. Let's compute that:

139.6 * 139.6:

Let me compute 140 * 140 = 19600, but since 139.6 is 0.4 less than 140, the square would be (140 - 0.4)^2 = 140² - 2*140*0.4 + 0.4² = 19600 - 112 + 0.16 = 19488.16

Similarly, 105.7 squared: 105.7 * 105.7. Let's compute 100*100=10000, 5.7*100=570, 100*5.7=570, 5.7*5.7≈32.49. So total is 10000 + 570 + 570 + 32.49 ≈ 11172.49

So, M_pi² - m_mu² = 19488.16 - 11172.49 ≈ 8315.67

Then, p = 8315.67 / (2 * 139.6) ≈ 8315.67 / 279.2 ≈ Let's compute that.

Divide 8315.67 by 279.2:

279.2 * 29 = 279.2 * 30 = 8376, minus 279.2 = 8376 - 279.2 = 8096.8

So 29 times 279.2 is 8096.8. Subtract that from 8315.67: 8315.67 - 8096.8 = 218.87

Now, 218.87 / 279.2 ≈ 0.783

So total p ≈ 29.783 MeV/c

Wait, but units here. Since we're using natural units where c=1, the momentum p is in MeV/c. But since c=1, the unit for energy is MeV and for momentum is MeV/c. But when we compute kinetic energy, which is energy minus rest mass, so for the muon, K_mu = sqrt(p² + m_mu²) - m_mu.

But let's compute p numerically. Let's get more precise.

First, compute M_pi² - m_mu²:

139.6 * 139.6 = (139 + 0.6)^2 = 139² + 2*139*0.6 + 0.6² = 19321 + 166.8 + 0.36 = 19321 + 166.8 is 19487.8 + 0.36 = 19488.16

105.7 * 105.7: Let's compute it step by step.

105 * 105 = 11025

105 * 0.7 = 73.5

0.7 * 105 = 73.5

0.7 * 0.7 = 0.49

So, (105 + 0.7)^2 = 105² + 2*105*0.7 + 0.7² = 11025 + 147 + 0.49 = 11172.49

So, M_pi² - m_mu² = 19488.16 - 11172.49 = 8315.67

Then, p = 8315.67 / (2 * 139.6) = 8315.67 / 279.2

Let me compute 279.2 * 29 = 279.2 * 30 = 8376 - 279.2 = 8096.8

8315.67 - 8096.8 = 218.87

Now, 218.87 / 279.2 ≈ Let's do this division:

279.2 * 0.78 = 279.2 * 0.7 = 195.44; 279.2 * 0.08 = 22.336. Total 195.44 + 22.336 = 217.776

So, 0.78 gives 217.776, which is very close to 218.87. The difference is 218.87 - 217.776 = 1.094

So, 1.094 / 279.2 ≈ 0.00392

So, total p ≈ 29.78 + 0.00392 ≈ 29.7839 MeV/c

So, p ≈ 29.784 MeV/c

Now, let's compute the kinetic energies.

First, for the neutrino: K_nu = p = 29.784 MeV

Wait, but wait. Wait, the neutrino's energy is p, and its rest mass is negligible, so K_nu = p. So, K_nu ≈ 29.784 MeV

For the muon: K_mu = sqrt(p² + m_mu²) - m_mu

We can compute sqrt(p² + m_mu²):

p² = (29.784)^2 ≈ 887.05 (since 30²=900, so 29.784² is a bit less than 900)

m_mu² = (105.7)^2 = 11172.49

So, p² + m_mu² ≈ 887.05 + 11172.49 ≈ 12059.54

sqrt(12059.54) ≈ Let's see. 109.8² = 12056.04 (since 100²=10000, 10²=100, 0.8²=0.64, cross terms: 2*100*10*0.8=1600, so (100 + 10 + 0.8)^2 ≈ 10000 + 2000 + 160 + 100 + 16 + 0.64 = Wait, maybe a better way. Alternatively, 109.8^2:

100^2 = 10000

2*100*9.8 = 1960

9.8^2 = 96.04

So total 10000 + 1960 + 96.04 = 12056.04

So sqrt(12059.54) is slightly more than 109.8. Let's compute 109.8^2 = 12056.04

Difference: 12059.54 - 12056.04 = 3.5

So, sqrt(12059.54) ≈ 109.8 + (3.5)/(2*109.8) ≈ 109.8 + 3.5/219.6 ≈ 109.8 + 0.0159 ≈ 109.816

So, sqrt(p² + m_mu²) ≈ 109.816 MeV

Therefore, K_mu = 109.816 - 105.7 ≈ 4.116 MeV

So, K_mu ≈ 4.116 MeV

And K_nu ≈ 29.784 MeV

Looking at the options, option D is 4.12 MeV, 29.8 MeV. That's very close. So the answer should be D.

Wait, but let me double-check my calculations to be sure.

First, M_pi = 139.6, m_mu = 105.7

p = (139.6² - 105.7²)/(2*139.6) = (19488.16 - 11172.49)/279.2 = 8315.67/279.2 ≈ 29.784 MeV/c

Then, E_mu = sqrt(p² + m_mu²) = sqrt(29.784² + 105.7²) ≈ sqrt(887.05 + 11172.49) = sqrt(12059.54) ≈ 109.816 MeV

So K_mu = E_mu - m_mu = 109.816 - 105.7 ≈ 4.116 MeV

K_nu = p ≈ 29.784 MeV

So the pair is approximately (4.12, 29.8), which is option D.

But wait, the options are:

A. 7.2, 32.8

B. 2.84, 26.8

C. 3.52, 20.8

D. 4.12, 29.8

So D matches our calculation.

But wait, let me check another way. Maybe using the formula for decay of a particle at rest into two particles. The formula for the kinetic energy of the muon is (M_pi² - m_mu² - m_nu²)/(2 M_pi). But since neutrino is massless (m_nu ≈ 0), it's (M_pi² - m_mu²)/(2 M_pi).

Wait, that's actually the same as we derived earlier. Because when you compute p, it's (M_pi² - m_mu²)/(2 M_pi). Then, the kinetic energy of the muon is sqrt(p² + m_mu²) - m_mu. Let's compute that again.

Alternatively, since the neutrino is massless, its energy is p, which we found to be 29.784 MeV, and the muon's energy is 109.816 MeV, so kinetic energy is 4.116 MeV. So option D is correct.

Alternatively, another approach is to use the formula for kinetic energy in two-body decay. For a particle of mass M decaying into two particles of mass m1 and m2 (with m2 = 0 for neutrino), the kinetic energy of m1 is (M² - m1²)/(2 M) - m1. Wait, no. Wait, the total energy is M, so the muon's energy is (M² - m1²)/(2 M) + m1/2? Wait, maybe not. Let me recall the formula.

In the rest frame of the original particle, the energies of the decay products are given by:

E1 = (M² + m1² - m2²)/(2 M)

E2 = (M² - m1² + m2²)/(2 M)

But since m2 is the neutrino and we can take m2=0, this simplifies to:

E1 = (M² + m1²)/(2 M)

E2 = (M² - m1²)/(2 M)

Then, the kinetic energy of each particle is E1 - m1 and E2 - 0.

So for the muon:

K_mu = E1 - m_mu = (M² + m_mu²)/(2 M) - m_mu = (M² + m_mu² - 2 M m_mu)/(2 M) = (M - m_mu)²/(2 M)

Similarly, K_nu = E2 = (M² - m_mu²)/(2 M)

Wait, let's compute K_mu using this formula.

K_mu = (M_pi - m_mu)^2 / (2 M_pi)

M_pi = 139.6, m_mu = 105.7

So, M_pi - m_mu = 139.6 - 105.7 = 33.9 MeV

Then, K_mu = (33.9)^2 / (2 * 139.6) = 1149.21 / 279.2 ≈ 4.118 MeV

Which is approximately 4.12 MeV, which matches our previous calculation.

Similarly, K_nu = (M_pi² - m_mu²)/(2 M_pi) = (139.6² - 105.7²)/(2*139.6) = 8315.67 / 279.2 ≈ 29.784 MeV

So again, K_nu ≈ 29.8 MeV

Therefore, the pair is (4.12, 29.8), which is option D.

So the correct answer is D.
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