University Physics III · topic UPIII-11.05
The decay law, activity, and decay chains
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.
The decay law, activity, and decay chains: predict a finite change
The decay law says every nucleus of a given unstable nuclide has the same fixed probability λ per unit time of decaying, whatever its age, so the expected number surviving is N₀e^(−λt) and the sample's activity, its decays per second, is A = λN. Use the supplied topic model Y_out = (1-2^(-(1/r-1)*t))/(1-r), where Y_out is A₂/A₁ AT CURSOR (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. t is Cursor time t, measured in T₁. Keep r (Half-life ratio T₂/T₁) = 0.3 fixed. For this exercise use only 0.6375 T₁ ≤ t ≤ 5.3625 T₁ 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 t = 1.677 T₁ and 2.3385 T₁. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δt across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
A₂/A₁ AT CURSOR against t
Predictions from the supplied The decay law, activity, and decay chains 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): A₂/A₁ AT CURSOR
Horizontal axis (x): t: Cursor time t / T₁
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| t: Cursor time t / T₁ | A₂/A₁ AT CURSOR |
|---|---|
| 0.6375 | 0.9190999907610615 |
| 1.8187499999999999 | 1.3531654264619575 |
| 3 | 1.4174107142857144 |
| 4.1812499999999995 | 1.4269195502078666 |
| 5.3625 | 1.4283269368607594 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of t separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At t = 1.677 T₁, substitute into Y_out = (1-2^(-(1/r-1)*t))/(1-r): Y₀ = 1.33374.
- At t = 2.3385 T₁, the same fixed-parameter model gives Y₁ = 1.39604.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.0623021; the input change is Δt = 0.6615 T₁.
- Average sensitivity is ΔY_out/Δt = 0.094183 1 per T₁. 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 = 1.33374, Y₁ = 1.39604; ΔY_out = 0.0623021; average sensitivity = 0.094183 1 per T₁.
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.
The decay law, activity, and decay chains: infer an input with uncertainty
The decay law says every nucleus of a given unstable nuclide has the same fixed probability λ per unit time of decaying, whatever its age, so the expected number surviving is N₀e^(−λt) and the sample's activity, its decays per second, is A = λN. Use the supplied topic model Y_out = (1-2^(-(1/r-1)*t))/(1-r), where Y_out is A₂/A₁ AT CURSOR (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. t is Cursor time t, measured in T₁. Keep r (Half-life ratio T₂/T₁) = 0.3 fixed. For this exercise use only 0.6375 T₁ ≤ t ≤ 5.3625 T₁ 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 = 1.4237562, with a bounded reading uncertainty ±0.00068561. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t 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 T₁, 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.
A₂/A₁ AT CURSOR against t
Predictions from the supplied The decay law, activity, and decay chains 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): A₂/A₁ AT CURSOR
Horizontal axis (x): t: Cursor time t / T₁
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| t: Cursor time t / T₁ | A₂/A₁ AT CURSOR |
|---|---|
| 0.6375 | 0.9190999907610615 |
| 1.8187499999999999 | 1.3531654264619575 |
| 3 | 1.4174107142857144 |
| 4.1812499999999995 | 1.4269195502078666 |
| 5.3625 | 1.4283269368607594 |
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(t) = 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(0.6375) = 0.9191 and Y_out(5.3625) = 1.42833. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives t_line = 0.6375 + (1.42376 − 0.9191)(5.3625 − 0.6375)/(1.42833 − 0.9191) = 5.32009 T₁. 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 t = 3.51975 T₁; substituting back gives Y_out = 1.4237562.
- The interpolation error in the input is |t_line − t| = 1.80034 T₁. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 1.4230706 and 1.4244418. Sort the roots even if the response decreases: 3.43745 T₁ ≤ t ≤ 3.61472 T₁.
- The lower and upper input deviations are 0.082306 T₁ and 0.0949696 T₁. 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 t ≈ 3.51975 T₁; compatible interval [3.43745, 3.61472] T₁. Endpoint interpolation gives 5.32009 T₁.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.