University Physics I · topic UPI-13.03
Gravitational potential energy
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.
Gravitational potential energy: predict a finite change
Gravitational potential energy U(r) = −GMm/r is the energy of two masses a distance r apart, measured from zero at infinite separation: it equals minus the work gravity does as the masses are brought together from infinitely far apart. Use the supplied topic model Y_out = 62.57*(r2-r1)/r1^2, where Y_out is CONSTANT-FORCE GUESS (MJ kg⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. r1 is Start radius r₁, measured in Earth radii. Keep r2 (End radius r₂) = 4 Earth radii fixed. For this exercise use only 1.2125 Earth radii ≤ r1 ≤ 2.7875 Earth radii 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 r1 = 1.559 Earth radii and 1.7795 Earth radii. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δr1 across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
CONSTANT-FORCE GUESS against r1
Predictions from the supplied Gravitational potential energy 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): CONSTANT-FORCE GUESS / MJ kg⁻¹
Horizontal axis (x): r1: Start radius r₁ / Earth radii
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r1: Start radius r₁ / Earth radii | CONSTANT-FORCE GUESS / MJ kg⁻¹ |
|---|---|
| 1.2125 | 118.63628440854504 |
| 1.6062499999999997 | 58.052197610864695 |
| 1.9999999999999998 | 31.285000000000007 |
| 2.39375 | 17.539681912072485 |
| 2.7874999999999996 | 9.763783707695717 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of r1 separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At r1 = 1.559 Earth radii, substitute into Y_out = 62.57*(r2-r1)/r1^2: Y₀ = 62.8408 MJ kg⁻¹.
- At r1 = 1.7795 Earth radii, the same fixed-parameter model gives Y₁ = 43.8754 MJ kg⁻¹.
- Signed change: ΔY_out = Y₁ − Y₀ = -18.9654 MJ kg⁻¹; the input change is Δr1 = 0.2205 Earth radii.
- Average sensitivity is ΔY_out/Δr1 = -86.0109 MJ kg⁻¹ per Earth radii. 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 = 62.8408 MJ kg⁻¹, Y₁ = 43.8754 MJ kg⁻¹; ΔY_out = -18.9654 MJ kg⁻¹; average sensitivity = -86.0109 MJ kg⁻¹ per Earth radii.
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.
Gravitational potential energy: infer an input with uncertainty
Gravitational potential energy U(r) = −GMm/r is the energy of two masses a distance r apart, measured from zero at infinite separation: it equals minus the work gravity does as the masses are brought together from infinitely far apart. Use the supplied topic model Y_out = 62.57*(r2-r1)/r1^2, where Y_out is CONSTANT-FORCE GUESS (MJ kg⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. r1 is Start radius r₁, measured in Earth radii. Keep r2 (End radius r₂) = 4 Earth radii fixed. For this exercise use only 1.2125 Earth radii ≤ r1 ≤ 2.7875 Earth radii 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 = 24.200587 MJ kg⁻¹, with a bounded reading uncertainty ±2.1655 MJ kg⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r1 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575 Earth radii, 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.
CONSTANT-FORCE GUESS against r1
Predictions from the supplied Gravitational potential energy 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): CONSTANT-FORCE GUESS / MJ kg⁻¹
Horizontal axis (x): r1: Start radius r₁ / Earth radii
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r1: Start radius r₁ / Earth radii | CONSTANT-FORCE GUESS / MJ kg⁻¹ |
|---|---|
| 1.2125 | 118.63628440854504 |
| 1.6062499999999997 | 58.052197610864695 |
| 1.9999999999999998 | 31.285000000000007 |
| 2.39375 | 17.539681912072485 |
| 2.7874999999999996 | 9.763783707695717 |
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(r1) = 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(1.2125) = 118.636 MJ kg⁻¹ and Y_out(2.7875) = 9.76378 MJ kg⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives r1_line = 1.2125 + (24.2006 − 118.636)(2.7875 − 1.2125)/(9.76378 − 118.636) = 2.57865 Earth radii. 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 r1 = 2.17325 Earth radii; substituting back gives Y_out = 24.200587 MJ kg⁻¹.
- The interpolation error in the input is |r1_line − r1| = 0.405401 Earth radii. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 22.035087 MJ kg⁻¹ and 26.366087 MJ kg⁻¹. Sort the roots even if the response decreases: 2.11502 Earth radii ≤ r1 ≤ 2.23727 Earth radii.
- The lower and upper input deviations are 0.0582327 Earth radii and 0.0640206 Earth radii. 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 r1 ≈ 2.17325 Earth radii; compatible interval [2.11502, 2.23727] Earth radii. Endpoint interpolation gives 2.57865 Earth radii.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.