Cosmic rays are high-energy particles that come from space. Most of these particles are protons. There are other particles in cosmic rays, including atomic nuclei. Table 1 gives the data for one particular nucleus X. Table 1 | Mass / kg | 8.02 × 10^{-26} | | Specific charge / C kg^{-1} | 4.39 × 10^{7} | | Kinetic energy / MeV | 215 | 0 1 1. Determine the number of neutrons in nucleus X. number of neutrons = 0 1 2. Calculate the speed of X. Ignore relativistic effects. 0 1 3. A pion (π^{−}) and a kaon (K^{+}) are produced when cosmic rays interact with the upper atmosphere. The π^{−} decays to produce a positron and an electron neutrino. 0 1 4. Show how the conservation laws apply to this decay. 0 1 5. The K^{+} decays to produce an anti-muon and a muon neutrino. Explain how strangeness applies in this decay. 0 1 6. Write an equation for a K^{+} decay that involves only hadrons.

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Cosmic rays are high-energy particles that come from space - AQA - A-Level Physics - Question 1 - 2021 - Paper 1

Question 1

Cosmic rays are high-energy particles that come from space. Most of these particles are protons. There are other particles in cosmic rays, including atomic nuclei. ... show full transcript

Worked Solution & Example Answer:Cosmic rays are high-energy particles that come from space - AQA - A-Level Physics - Question 1 - 2021 - Paper 1

Step 1

Determine the number of neutrons in nucleus X.

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Answer

To find the number of neutrons, we can use the specific charge and the mass of the nucleus. The specific charge is defined as the charge divided by the mass:

$ext{Specific Charge} = \frac{Q}{m}$

Where:

Specific Charge = $4.39 \times 10^{7} \text{ C kg}^{-1}$
Mass = $8.02 \times 10^{-26} \text{ kg}$

Rearranging this gives:

$Q = ext{Specific Charge} \times m = (4.39 \times 10^{7}) \times (8.02 \times 10^{-26}) \approx 3.52 \times 10^{-18} \text{ C}$

Knowing the charge of a proton is approximately $1.6 \times 10^{-19} \text{ C}$ we can find the number of protons:

$ext{Number of Protons} = \frac{3.52 \times 10^{-18} \text{ C}}{1.6 \times 10^{-19} \text{ C}} \approx 22$

Using the mass of the nucleus (relative to the atomic mass unit where 1 atomic mass unit $ext{amu} \approx 1.66 \times 10^{-27} \text{ kg}$ ), the mass number A can be determined:

$A = \frac{8.02 \times 10^{-26} \text{ kg}}{1.66 \times 10^{-27} \text{ kg/amu}} \approx 48$

Thus, the number of neutrons can be calculated as:

$\text{Number of Neutrons} = A - Z = 48 - 22 = 26$

Step 2

Calculate the speed of X.

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Answer

To calculate the speed of nucleus X, we will use its kinetic energy. The kinetic energy (KE) of an object is given by:

$KE = \frac{1}{2}mv^{2}$

Where:

KE = $215 \text{ MeV} = 215 \times 1.6 \times 10^{-13} \text{ J} \approx 3.44 \times 10^{-11} \text{ J}$
m = $8.02 \times 10^{-26} \text{ kg}$

Rearranging the formula for speed gives:

$v = \sqrt{\frac{2 \times KE}{m}}$

Substituting the values:

$v = \sqrt{\frac{2 \times (3.44 \times 10^{-11})}{8.02 \times 10^{-26}}} \approx \sqrt{8.578 \times 10^{14}} \approx 9.27 \times 10^{7} \text{ m/s}$

Step 3

Show how the conservation laws apply to this decay.

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Answer

In the decay of the pion ( $π^{-} \to e^{-} + ar{ u}_e$ ), several conservation laws must be satisfied:

Conservation of Charge: The initial charge of the pion is -1, and the final charges of the products are -1 (electron) + 0 (neutrino) = -1.
Conservation of Lepton Number: The initial lepton number is 0 for the pion, and after the decay, we have +1 for the electron and -1 for the electron neutrino, meaning overall lepton number is conserved.
Conservation of Energy and Momentum: The energy and momentum before and after the decay must also be conserved, balanced by the initial energy of the pion and the energy of the emitted particles.

Step 4

Explain how strangeness applies in this decay.

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Answer

In the decay of the K^{+} meson to an anti-muon ( $μ^{-}$ ) and a muon neutrino ( $ar{ u}_{μ}$ ), strangeness is a key concept. Strangeness is a quantum number related to the presence of strange quarks in mesons. The K^{+} meson has a strangeness of +1 due to the presence of a strange quark, while the final state products (anti-muon and neutrino) have strangeness of 0.

Since strangeness is not conserved in strong interactions, K mesons can decay via the weak interaction, where strangeness can change (in this case, it changes from +1 to 0), allowing this decay to proceed.

Step 5

Write an equation for a K^{+} decay that involves only hadrons.

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Answer

An example of a K^{+} decay involving only hadrons is:

$K^{+} \to \pi^{+} + \pi^{0}$

In this decay process, a K^{+} meson decays into a positive pion and a neutral pion, thereby conserving baryon number, lepton number, and strangeness.

Cosmic rays are high-energy particles that come from space - AQA - A-Level Physics - Question 1 - 2021 - Paper 1

Worked Solution & Example Answer:Cosmic rays are high-energy particles that come from space - AQA - A-Level Physics - Question 1 - 2021 - Paper 1

Determine the number of neutrons in nucleus X.

Calculate the speed of X.

Show how the conservation laws apply to this decay.

Explain how strangeness applies in this decay.

Write an equation for a K^{+} decay that involves only hadrons.

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