University Physics V · topic UPV-14.09
Reading fission and fusion off the B/A curve
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.
Fission and fusion from the B/A curve: predict a finite change
Reading fission and fusion off the B/A curve means computing a reaction's Q value as the change in total binding energy, ΣB(final) − ΣB(initial), with every nucleus contributing its own B/A multiplied by its own mass number. Use the supplied topic model Y_out = f*m*(15.75-17.8/(f*m)^0.33333-0.24618*(f*m)^0.561)+(1-f)*m*(15.75-17.8/((1-f)*m)^0.33333-0.24618*((1-f)*m)^0.561)-m*(15.75-17.8/m^0.33333-0.24618*m^0.561), where Y_out is Q RELEASED (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. f is Heavy-fragment mass fraction, dimensionless. Keep m (Parent mass number A) = 236 fixed. For this exercise use only 0.526563 ≤ f ≤ 0.723437 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 f = 0.569875 and 0.597438. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δf across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
Q RELEASED against f
Predictions from the supplied Fission and fusion from the B/A curve 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): Q RELEASED / MeV
Horizontal axis (x): f: Heavy-fragment mass fraction
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| f: Heavy-fragment mass fraction | Q RELEASED / MeV |
|---|---|
| 0.5265625 | 223.82061252655035 |
| 0.57578125 | 218.29216495579885 |
| 0.625 | 207.459984745145 |
| 0.67421875 | 191.35598595498777 |
| 0.7234375 | 170.04300005591767 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of f separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At f = 0.569875, substitute into Y_out = f*m*(15.75-17.8/(f*m)^0.33333-0.24618*(f*m)^0.561)+(1-f)*m*(15.75-17.8/((1-f)*m)^0.33333-0.24618*((1-f)*m)^0.561)-m*(15.75-17.8/m^0.33333-0.24618*m^0.561): Y₀ = 219.236 MeV.
- At f = 0.597438, the same fixed-parameter model gives Y₁ = 214.178 MeV.
- Signed change: ΔY_out = Y₁ − Y₀ = -5.05797 MeV; the input change is Δf = 0.0275625.
- Average sensitivity is ΔY_out/Δf = -183.509 MeV per unit input. Its negative sign means that this response decreases 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 = 219.236 MeV, Y₁ = 214.178 MeV; ΔY_out = -5.05797 MeV; average sensitivity = -183.509 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.
Fission and fusion from the B/A curve: infer an input with uncertainty
Reading fission and fusion off the B/A curve means computing a reaction's Q value as the change in total binding energy, ΣB(final) − ΣB(initial), with every nucleus contributing its own B/A multiplied by its own mass number. Use the supplied topic model Y_out = f*m*(15.75-17.8/(f*m)^0.33333-0.24618*(f*m)^0.561)+(1-f)*m*(15.75-17.8/((1-f)*m)^0.33333-0.24618*((1-f)*m)^0.561)-m*(15.75-17.8/m^0.33333-0.24618*m^0.561), where Y_out is Q RELEASED (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. f is Heavy-fragment mass fraction, dimensionless. Keep m (Parent mass number A) = 236 fixed. For this exercise use only 0.526563 ≤ f ≤ 0.723437 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 = 201.02092 MeV, with a bounded reading uncertainty ±3.42 MeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown f by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000196875, 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.
Q RELEASED against f
Predictions from the supplied Fission and fusion from the B/A curve 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): Q RELEASED / MeV
Horizontal axis (x): f: Heavy-fragment mass fraction
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| f: Heavy-fragment mass fraction | Q RELEASED / MeV |
|---|---|
| 0.5265625 | 223.82061252655035 |
| 0.57578125 | 218.29216495579885 |
| 0.625 | 207.459984745145 |
| 0.67421875 | 191.35598595498777 |
| 0.7234375 | 170.04300005591767 |
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(f) = 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(0.526563) = 223.821 MeV and Y_out(0.723437) = 170.043 MeV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives f_line = 0.526563 + (201.021 − 223.821)(0.723437 − 0.526563)/(170.043 − 223.821) = 0.61003. 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 f = 0.646656; substituting back gives Y_out = 201.02092 MeV.
- The interpolation error in the input is |f_line − f| = 0.0366261. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 197.60092 MeV and 204.44092 MeV. Sort the roots even if the response decreases: 0.635582 ≤ f ≤ 0.656957.
- The lower and upper input deviations are 0.0110741 and 0.0103012. 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 f ≈ 0.646656; compatible interval [0.635582, 0.656957]. Endpoint interpolation gives 0.61003.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.