University Physics III · topic UPIII-11.09
Quarks, leptons, and the Standard Model
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Quarks, leptons, and the Standard Model: predict a finite change
The Standard Model sorts matter into six quarks and six leptons, describes the strong, electromagnetic and weak interactions as the exchange of carrier bosons, and supplies the conserved numbers that decide which reactions can occur. Use the supplied topic model Y_out = 1000*10^m, where Y_out is REST ENERGY Mc² (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. m is log₁₀ of rest energy Mc² in GeV, dimensionless. Keep a (Coupling α at short range) = 0.034 fixed. For this exercise use only -3.3625 ≤ m ≤ 1.3625 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at m = -2.323 and -1.6615. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δm across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
REST ENERGY Mc² against m
Predictions from the supplied Quarks, leptons, and the Standard Model model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): REST ENERGY Mc² / MeV
Horizontal axis (x): m: log₁₀ of rest energy Mc² in GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: log₁₀ of rest energy Mc² in GeV | REST ENERGY Mc² / MeV |
|---|---|
| -3.3625 | 0.43401026364474404 |
| -2.18125 | 6.587945534419239 |
| -1 | 100 |
| 0.1812499999999999 | 1517.9239032493838 |
| 1.3624999999999998 | 23040.929760558443 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of m separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At m = -2.323, substitute into Y_out = 1000*10^m: Y₀ = 4.75335 MeV.
- At m = -1.6615, the same fixed-parameter model gives Y₁ = 21.8022 MeV.
- Signed change: ΔY_out = Y₁ − Y₀ = 17.0488 MeV; the input change is Δm = 0.6615.
- Average sensitivity is ΔY_out/Δm = 25.773 MeV per unit input. Its positive sign means that this response increases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 4.75335 MeV, Y₁ = 21.8022 MeV; ΔY_out = 17.0488 MeV; average sensitivity = 25.773 MeV per unit input.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Quarks, leptons, and the Standard Model: infer an input with uncertainty
The Standard Model sorts matter into six quarks and six leptons, describes the strong, electromagnetic and weak interactions as the exchange of carrier bosons, and supplies the conserved numbers that decide which reactions can occur. Use the supplied topic model Y_out = 1000*10^m, where Y_out is REST ENERGY Mc² (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. m is log₁₀ of rest energy Mc² in GeV, dimensionless. Keep a (Coupling α at short range) = 0.034 fixed. For this exercise use only -3.3625 ≤ m ≤ 1.3625 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 330.94056 MeV, with a bounded reading uncertainty ±49.576 MeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown m by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0004725, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
REST ENERGY Mc² against m
Predictions from the supplied Quarks, leptons, and the Standard Model model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): REST ENERGY Mc² / MeV
Horizontal axis (x): m: log₁₀ of rest energy Mc² in GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: log₁₀ of rest energy Mc² in GeV | REST ENERGY Mc² / MeV |
|---|---|
| -3.3625 | 0.43401026364474404 |
| -2.18125 | 6.587945534419239 |
| -1 | 100 |
| 0.1812499999999999 | 1517.9239032493838 |
| 1.3624999999999998 | 23040.929760558443 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(m) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(-3.3625) = 0.43401 MeV and Y_out(1.3625) = 23040.9 MeV. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives m_line = -3.3625 + (330.941 − 0.43401)(1.3625 − -3.3625)/(23040.9 − 0.43401) = -3.29472. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives m = -0.48025; substituting back gives Y_out = 330.94056 MeV.
- The interpolation error in the input is |m_line − m| = 2.81447. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 281.36456 MeV and 380.51656 MeV. Sort the roots even if the response decreases: -0.550731 ≤ m ≤ -0.419626.
- The lower and upper input deviations are 0.0704806 and 0.0606236. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer m ≈ -0.48025; compatible interval [-0.550731, -0.419626]. Endpoint interpolation gives -3.29472.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.