University Physics IV · topic UPIV-14.01
Relativistic kinematics and invariant mass
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.
Relativistic kinematics and invariant mass: predict a finite change
The invariant mass of a system is M = √(E_total² − |p_total c|²)/c², assembled from the total energy and the total momentum, and every inertial observer computes the same value for it. Use the supplied topic model Y_out = (4*eb*eb-2*mt*mt)/(2*max(mt,0.05)), where Y_out is TARGET BEAM TO MATCH (GeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. mt is Target rest mass m, measured in GeV/c². Keep eb (Beam energy per beam E) = 46 GeV fixed. For this exercise use only 0.301875 GeV/c² ≤ mt ≤ 1.79812 GeV/c² 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 mt = 0.63105 GeV/c² and 0.840525 GeV/c². Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δmt across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
TARGET BEAM TO MATCH against mt
Predictions from the supplied Relativistic kinematics and invariant mass 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): TARGET BEAM TO MATCH / GeV
Horizontal axis (x): mt: Target rest mass m / GeV/c²
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| mt: Target rest mass m / GeV/c² | TARGET BEAM TO MATCH / GeV |
|---|---|
| 0.301875 | 14018.745744047617 |
| 0.6759375 | 6260.257950618355 |
| 1.0499999999999998 | 4029.426190476191 |
| 1.4240624999999998 | 2970.355617113781 |
| 1.7981249999999998 | 2351.7646139641993 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of mt separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At mt = 0.63105 GeV/c², substitute into Y_out = (4*eb*eb-2*mt*mt)/(2*max(mt,0.05)): Y₀ = 6705.65 GeV.
- At mt = 0.840525 GeV/c², the same fixed-parameter model gives Y₁ = 5034.11 GeV.
- Signed change: ΔY_out = Y₁ − Y₀ = -1671.54 GeV; the input change is Δmt = 0.209475 GeV/c².
- Average sensitivity is ΔY_out/Δmt = -7979.68 GeV per GeV/c². 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 = 6705.65 GeV, Y₁ = 5034.11 GeV; ΔY_out = -1671.54 GeV; average sensitivity = -7979.68 GeV per GeV/c².
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.
Relativistic kinematics and invariant mass: infer an input with uncertainty
The invariant mass of a system is M = √(E_total² − |p_total c|²)/c², assembled from the total energy and the total momentum, and every inertial observer computes the same value for it. Use the supplied topic model Y_out = (4*eb*eb-2*mt*mt)/(2*max(mt,0.05)), where Y_out is TARGET BEAM TO MATCH (GeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. mt is Target rest mass m, measured in GeV/c². Keep eb (Beam energy per beam E) = 46 GeV fixed. For this exercise use only 0.301875 GeV/c² ≤ mt ≤ 1.79812 GeV/c² 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 = 3483.0959 GeV, with a bounded reading uncertainty ±169.7 GeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown mt by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000149625 GeV/c², 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.
TARGET BEAM TO MATCH against mt
Predictions from the supplied Relativistic kinematics and invariant mass 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): TARGET BEAM TO MATCH / GeV
Horizontal axis (x): mt: Target rest mass m / GeV/c²
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| mt: Target rest mass m / GeV/c² | TARGET BEAM TO MATCH / GeV |
|---|---|
| 0.301875 | 14018.745744047617 |
| 0.6759375 | 6260.257950618355 |
| 1.0499999999999998 | 4029.426190476191 |
| 1.4240624999999998 | 2970.355617113781 |
| 1.7981249999999998 | 2351.7646139641993 |
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(mt) = 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(0.301875) = 14018.7 GeV and Y_out(1.79812) = 2351.76 GeV. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives mt_line = 0.301875 + (3483.1 − 14018.7)(1.79812 − 0.301875)/(2351.76 − 14018.7) = 1.65304 GeV/c². 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 mt = 1.21459 GeV/c²; substituting back gives Y_out = 3483.0959 GeV.
- The interpolation error in the input is |mt_line − mt| = 0.438448 GeV/c². This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 3313.3959 GeV and 3652.7959 GeV. Sort the roots even if the response decreases: 1.1582 GeV/c² ≤ mt ≤ 1.27675 GeV/c².
- The lower and upper input deviations are 0.0563901 GeV/c² and 0.06216 GeV/c². 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 mt ≈ 1.21459 GeV/c²; compatible interval [1.1582, 1.27675] GeV/c². Endpoint interpolation gives 1.65304 GeV/c².
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.