Gravity is in the curvature, not just the distance fallen
A falling object travels farther in each successive equal time interval. That changing motion contains information about gravitational acceleration. But the answer is not simply the distance divided by time: initial position, initial velocity, the clock scale, the length scale and the reference against which motion is observed all enter the inference.
This guide develops an ideal ballistic gravimeter from a quadratic trajectory. You will recover acceleration with a three-position check, change the time origin, introduce calibration errors, and distinguish a measured trajectory from gravity quoted at a benchmark. Later examples ask which uncertainties cancel and which remain after many repeated drops.
The calculations are original fictional fixtures, not instrument operating procedures, real gravity observations or claims about a laboratory’s performance. An actual absolute gravimeter needs its own validated observation model, calibration evidence and environmental corrections. The word “absolute” identifies a measurement approach; it does not mean uncertainty-free.
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Choose your reading route
Route 1 · Chapters 1–2: Identify the observable
Route 2 · Chapters 3–5: Recover and test curvature
Recover acceleration with one original three-position fixture
Route 3 · Chapters 6–8: Follow uncertainty and calibration
Carry the position covariance into the acceleration estimate
Route 4 · Chapters 9–10: Define the reported gravity
Route 5 · Chapters 11–12: Transfer and assess evidence
Use repeated drops to test precision without overclaiming accuracy
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Identify the observable
Ask what the gravimeter observes before naming its output
Build the quadratic trajectory from constant acceleration
Recover and test curvature
Recover acceleration with one original three-position fixture
Change the origin without changing the acceleration
Understand what more observations add to the fit
Follow uncertainty and calibration
Carry the position covariance into the acceleration estimate
Derive how length and time scales bias the answer
Distinguish reference vibration from the falling body’s motion
Define the reported gravity
State the height at which the gravity value applies
Separate temporal gravity changes from instrument changes
CHAPTER 1 OF 12 · Identify the observable
1. Ask what the gravimeter observes before naming its output
A ballistic absolute gravimeter infers local gravitational acceleration from a tracked test body’s motion. INRIM describes its rise-and-fall instrument as using interferometry to observe vertical free motion. Its associated trajectory research describes fitting a motion model to time-position coordinates. Those are the specialist starting points; the following three-sample exercise is a deliberately simpler mathematical model.
Separate the observation from the estimate. A sequence of interference observations is processed into information about displacement and time. A fitted model then yields an acceleration parameter. Calling each raw observation a gravity value would skip the relationships that connect the instrument’s record to the quantity eventually reported. Those relationships are where calibration and modelling enter.
Absolute measurement contrasts with merely comparing a gravity-sensitive reading against another unknown gravity value. It still relies on realised units, reference behaviour, instrument models and corrections. A result expressed directly in metres per second squared can contain a systematic scale error. The unit label tells you the dimension of the answer, not whether its realisation was correct.
For our first model, choose a downward-positive vertical coordinate z in metres and time t in seconds. Assume constant downward acceleration g, no disturbing forces, and a suitable non-accelerating observational reference over the interval. These simplifications make a clean derivation possible. Keep them visible so that the derivation is not mistaken for a complete instrument model.
CHAPTER 2 OF 12 · Identify the observable
2. Build the quadratic trajectory from constant acceleration
If acceleration is constant, velocity changes linearly: v(t) = v0 + gt. Integrating once more gives z(t) = z0 + v0t + ½gt². The three parameters have different roles. Initial position sets the offset; initial velocity sets the linear part; acceleration supplies the curvature. A measurement that does not separate those roles can confuse release conditions with gravity.
The coefficient of t² is half the acceleration, not the acceleration itself. If a polynomial fit reports z = a + bt + ct², then this model gives g = 2c. A factor-of-two error can survive a beautifully smooth fit because the fitted curve is unchanged; the error occurs when translating its coefficient into the physical parameter.
Use dimensional checks. z0 has units of metres, v0t also has units of metres, and gt² does too. A velocity term cannot be added directly to an acceleration term without its corresponding time factor. These checks remain useful when a longer instrument model includes additional powers, sensitivities or corrections.
A fitted parabola alone does not prove that gravity was the only acceleration. Any unmodelled contribution with similar curvature can enter the same parameter. The clean equation is an explanation of identifiability under declared assumptions. Establishing that those assumptions describe an actual measurement requires external evidence about the reference, instrument, environment and trajectory.
CHAPTER 3 OF 12 · Recover and test curvature
3. Recover acceleration with one original three-position fixture
Take three mathematical time labels −T, 0 and +T, equally spaced around a chosen origin. Substituting them into the quadratic and forming z(+T) − 2z(0) + z(−T) cancels the initial position and initial velocity. The remaining term is gT². Therefore g = [z(+T) − 2z(0) + z(−T)]/T².
Choose T = 0.1 s, z0 = 1 m, v0 = 0.2 m/s and g = 9.8 m/s². The positions are z(−T) = 1.029 m, z(0) = 1.000 m and z(+T) = 1.069 m. Their second difference is 1.069 − 2.000 + 1.029 = 0.098 m. Dividing by 0.01 s² recovers 9.8 m/s².
The negative time label is simply an origin choice; it does not require observing before the experiment exists. One can relabel three observations around their middle epoch. What matters is equal spacing and the assumed quadratic behaviour. If the time gaps differ, this particular second-difference formula no longer has the same cancellation properties.
Inspect an independent scale check: acceleration times T² is 9.8 × 0.01 = 0.098 m, matching the observed curvature combination. The total change from the first position to the last is only 0.040 m, which reflects the linear term over the symmetric interval. Treating that total displacement as the curvature would infer the wrong quantity.
CHAPTER 4 OF 12 · Recover and test curvature
4. Change the origin without changing the acceleration
Let the old time be t = t′ + a. Substituting into z0 + v0t + ½gt² produces a new constant, a new linear coefficient and the same quadratic coefficient ½g. The new initial velocity is v0 + ga. Thus shifting the time origin changes the descriptive release parameters without changing constant acceleration.
A spatial origin change behaves even more simply. Adding a fixed number K to every position changes z0 but not velocity or acceleration. In the second difference, its contribution is K − 2K + K = 0. This is a useful implementation test: translating every position by the same amount should not alter the recovered g in this model.
A linear position drift a + bt also cancels from the ideal equally spaced second difference. That mathematical cancellation does not make all instrument drift harmless. A drift with quadratic content does not cancel, unequal time spacing changes the combination, and a real optical reference can affect the observations in ways not represented by an additive linear position term.
Try the checks in both directions. Shift the three fixture positions by +100 m and confirm the same second difference. Relabel their times as 0, 0.1 and 0.2 s and fit a quadratic with a new intercept and slope. If acceleration changes solely because of these redefinitions, examine the implementation before seeking a physical explanation.
CHAPTER 5 OF 12 · Recover and test curvature
5. Understand what more observations add to the fit
Three exact positions determine the three coefficients of the ideal quadratic. They leave no residual degrees of freedom with which to assess that curve’s adequacy. A perfect three-point fit is therefore a weak model check. It can reproduce three contaminated values exactly while reporting a biased acceleration.
Additional observations allow residual inspection and a weighted fit. The design matrix has columns 1, t and ½t² if the fitted parameters are z0, v0 and g. Stating those columns prevents confusion between acceleration and the raw quadratic coefficient. Observation weights and covariance determine how the data influence each parameter.
Temporal coverage matters as well as count. Many observations clustered into a very short interval can separate curvature poorly from noise. Observations distributed over a longer interval can improve leverage on curvature, but only if the constant-acceleration model and the treatment of other effects remain adequate. Extending an interval is not a universal route to a better physical result.
Use held-out checks where appropriate: fit a subset, predict another part, and examine whether residual patterns follow time or position. A residual curve can indicate missing model structure, but it does not uniquely identify the cause. The purpose is to discover which assumption needs further evidence, not to label every departure “gravity variation” by default.
Unequal intervals offer a further independent check. For successive times with positive gaps h1 and h2, form the first secant slope s1 = (z1 − z0)/h1 and the second s2 = (z2 − z1)/h2. For an exact quadratic, acceleration is 2(s2 − s1)/(h1 + h2). Using z = ½gt² at times 0, 0.1 and 0.3 s with g = 10 m/s² gives positions 0, 0.05 and 0.45 m. The slopes are 0.5 and 2 m/s, so 2 × 1.5/0.3 = 10 m/s². Treating those intervals as equal would discard essential timing information. This small fixture tests sampling logic separately from the symmetric example; real noisy observations still require an appropriate fit and uncertainty model.
CHAPTER 6 OF 12 · Follow uncertainty and calibration
6. Carry the position covariance into the acceleration estimate
The three-position estimate is a weighted combination with weights 1/T², −2/T² and 1/T². If each position has independent standard uncertainty σ_z, then Var(g) = (1 + 4 + 1)σ_z²/T⁴ = 6σ_z²/T⁴. The standard uncertainty is √6 σ_z/T². This expression covers only the stated position-noise model.
For illustration, choose σ_z = 0.000001 m and T = 0.1 s. The acceleration standard uncertainty is approximately 0.000245 m/s². That deliberately coarse fictional result is not a gravimeter performance specification. It makes the sensitivity of a short-interval second difference visible without implying that real instruments simply use three independent micrometre readings.
For correlated positions, use wᵀCw, where C is their covariance matrix and w is the weight vector. A perfectly shared position offset contributes zero because the weights sum to zero. Independent errors contribute differently. Equal printed uncertainty values do not tell you whether the errors are shared, independent or partly related.
The formula also treats observation times as exact. If timing is uncertain, it affects the estimated acceleration through the observation model and must be included appropriately. A small residual scatter in the positions cannot reveal a common time-scale error that distorts the entire trajectory consistently. Keep random observation noise separate from calibration and model uncertainty.
CHAPTER 7 OF 12 · Follow uncertainty and calibration
7. Derive how length and time scales bias the answer
Suppose the recorded time is t_m = (1 + ε)t_true, and the recorded position is z_m = (1 + κ)z_true, apart from additive terms that the fitted intercept and slope can absorb. Here ε and κ are dimensionless fractional scale errors. Substitute t_true = t_m/(1 + ε) into the true quadratic.
The fitted acceleration becomes g_fit = [(1 + κ)/(1 + ε)²]g_true. For small scale errors, the fractional acceleration error is approximately κ − 2ε. The factor of two arises because acceleration contains time squared. It is not an arbitrary calibration convention.
Use a deliberately exaggerated diagnostic value: if the time readings are 1% too large and the length scale is correct, the inferred acceleration is g_true/1.01², about 0.980296 times the truth. For g_true = 9.8 m/s², that is about 9.60690 m/s². The positions can still lie perfectly on a parabola in the recorded coordinates.
This is why repeated observations cannot establish every aspect of accuracy. A stable scale error can repeat faithfully. Calibration evidence must connect the realised length and time scales to appropriate standards and account for relevant instrument effects. Do not infer that evidence merely from a small spread among estimated drop values or from the use of a sophisticated optical system.
CHAPTER 8 OF 12 · Follow uncertainty and calibration
8. Distinguish reference vibration from the falling body’s motion
An instrument observes a relationship between the body and its measurement reference. If that reference moves, the observed trajectory can contain reference motion as well as test-body motion. In a simplified one-dimensional description, z_relative = z_body − z_reference. Differentiating twice gives a_relative = a_body − a_reference.
If the fictional body has downward acceleration 9.8 m/s² and the reference has downward acceleration 0.01 m/s² over the same ideal interval, the relative acceleration is 9.79 m/s². The relative record alone cannot decide whether the body accelerated less or the reference accelerated downward. An independent account of the reference is necessary.
This example explains the importance of reference stability and vibration treatment without prescribing an instrument’s isolation or correction method. INRIM identifies environmental effects and data processing as parts of its research programme. An actual implementation must use its validated model and records; the simple subtraction is not a substitute for that engineering.
A particularly useful evidence check asks which observation constrains the reference contribution and whether its uncertainty is carried into the final result. If a correction is estimated from the same data it modifies, dependence can matter. If a second instrument shares the same moving reference, agreement may not provide an independent test of the common error.
CHAPTER 9 OF 12 · Define the reported gravity
9. State the height at which the gravity value applies
Gravity can vary with height. A trajectory samples a range of positions, while a published station value may be required at a specified reference height. Those are not automatically the same location. State how the fitted parameter relates to height before comparing it with another instrument or benchmark value.
For a local teaching model, define upward height h and g(h) = g_ref + Γ(h − h_ref), where g is the positive downward magnitude and Γ is its vertical gradient. Choose the fictional gradient Γ = −0.000003 s⁻². This sign means that the gravity magnitude decreases as height increases in this model.
Suppose a properly defined fitted value applies at h_eff = h_ref + 0.5 m. Then g(h_eff) − g_ref = −0.0000015 m/s². In gravity units, 1 microgal is 0.00000001 m/s², so the difference is −150 microgals. Converting the measured value back down to h_ref would add 150 microgals under these assumptions.
Do not take h_eff to be the arithmetic mean of sampled heights without justification. The meaning of an effective measurement height depends on the trajectory model, sampling and estimator. The fixture assumes it has already been established appropriately. A real vertical-gradient treatment can be more elaborate, and a generic gradient should never be substituted for site-specific information merely because it gives a plausible number.
CHAPTER 10 OF 12 · Define the reported gravity
10. Separate temporal gravity changes from instrument changes
A repeated gravity measurement can differ because the physical gravity field changed, because the reference or instrument changed, or because the reductions differ. The same unit does not make two values comparable. Check location, height, epoch, environmental treatment and calibration before interpreting their difference.
In a fictional comparison, one result refers to h_ref and another to h_ref + 0.5 m under the previous gradient. A difference of −150 microgals would be explained by their reference heights alone. Calling it a temporal decrease in gravity would confuse a spatial change with a time change. Converting both to a compatible reference is part of defining the comparison.
Likewise, changing a correction convention can alter a reported value without changing the raw observation. Preserve the original estimate, each applied contribution, the adopted sign and the resulting quantity. Distinguish a contribution already present in the observation model from one added afterward. Two apparently different correction files can represent overlapping physics.
None of this means that genuine gravity changes should be removed indiscriminately. If the research question concerns a time-varying mass distribution, some environmental gravity variations may be the signal of interest. Define the question first, then decide which influences to retain, model or report separately. A flat series is not automatically the correct scientific output.
CHAPTER 11 OF 12 · Transfer and assess evidence
11. Use repeated drops to test precision without overclaiming accuracy
Under a simple independent, equal-variance error model, averaging n estimates reduces the variance of their mean by a factor of n. That rule concerns the independent part. A calibration bias common to all drops remains common. A slowly varying environmental disturbance can also make successive estimates dependent.
Write a toy estimate as g_i = g_true + b + e_i, with shared bias b and independent zero-mean errors e_i. The average is g_true + b + mean(e_i). Increasing n reduces the spread from mean(e_i), not b. If b is uncertain, its variance remains a separate contribution to the final uncertainty account.
A useful validation ladder therefore includes different kinds of evidence: repeated drops for short-term consistency, changed conditions for model robustness, calibration for scale realisation, and suitably independent comparisons for external agreement. No single rung replaces the others. Comparisons themselves need compatible references and an account of shared standards or reductions.
Avoid treating disagreement as an automatic fault in the less precise-looking instrument. Differences can arise from reference height, calibration, time, environment or model definitions. Work through those alternatives before assigning blame. The transferable habit is to ask what each check is capable of revealing and which shared error could pass through it unnoticed.
CHAPTER 12 OF 12 · Transfer and assess evidence
12. Report the inferred acceleration and its evidence together
A reusable gravity result identifies the instrument and observation interval, the location and reference height, the trajectory and estimation model, and the conventions applied. It distinguishes the fitted parameter from the final reduced value. Without that distinction, a later user may apply the same height or environmental contribution again.
State uncertainty with its meaning: standard or expanded uncertainty, any coverage information, and which calibration, observation, reference and model components were included. A narrow confidence interval conditional on a fixed scale is not the same as total measurement uncertainty. Explain shared components when comparing or averaging results.
Preserve enough intermediate information to reproduce the sign and scale of important adjustments. For the teaching fixture, that includes the three positions, equal time step, downward coordinate convention and factor of two between polynomial coefficient and acceleration. For an actual instrument, use the documentation and evidence appropriate to its much richer model.
The general lesson reaches beyond gravimetry. Curvature can distinguish acceleration from position and velocity, yet a mathematical distinction does not supply missing physical evidence. Keep the observation, the estimator, the units and the reference connected. Then a precise number becomes an interpretable measurement rather than an attractive output whose assumptions have disappeared.
Independent worked case and evidence checks
Work a different fixture before consulting the result. Take T = 0.2 s, z0 = 0.5 m, v0 = −0.1 m/s and constant downward g = 9.6 m/s². The three positions are 0.712 m, 0.500 m and 0.672 m. Their second difference is 0.384 m. Dividing by 0.04 s² returns 9.6 m/s².
Now add 7 m to every position. The second difference is unchanged. Add a linear position term of 0.03t m, with t in seconds, and it is unchanged again. Explain the cancellations using the weights, rather than relying on rounded calculator output. A quadratic added position term would not cancel, so this exercise does not license ignoring arbitrary reference motion.
For a calibration transfer check, let κ = 0.002 and ε = 0.001. The first-order expression κ − 2ε is zero, but the exact factor is 1.002/1.001², about 0.999999002. First-order cancellation is not exact cancellation. The discrepancy is small because it begins beyond first order, not because the two scales have been independently validated.
An evidence check completes the exercise: which part of a real gravity report would establish the clock scale, optical length scale, reference motion and reporting height? These are separate questions. The fictional arithmetic verifies the stated model, not any instrument’s accuracy or a site’s actual gravity.
Worked comparison
| Change to the ideal fixture | Effect on fitted g | Reason |
|---|---|---|
| Add a constant to all positions | None | Second-difference weights sum to zero |
| Add a linear position term | None for equal time spacing | Linear terms cancel |
| Stretch recorded times by 1 + ε | Divide g by (1 + ε)² | Acceleration contains time squared |
| Stretch recorded lengths by 1 + κ | Multiply g by 1 + κ | Position scale multiplies curvature |
| Repeat with the same unknown scale bias | Bias remains | Repeatability is not independent calibration |
Questions and practical checks
Why is distance divided by time squared not enough?
The initial velocity and position contribute to the observed displacement. The symmetric second difference removes those terms under the stated assumptions. More general fits estimate them alongside acceleration. The model’s time spacing and coefficient conventions must be preserved.
Does absolute gravity mean exact gravity?
No. Absolute gravimetry can infer acceleration using realised length and time scales rather than only a relative gravity response, but it still has calibration, observational, environmental and modelling uncertainty. A stable systematic contribution can remain after many repeated measurements.
Can I average the sampled heights to obtain an effective height?
Not without justification from the estimator and trajectory model. The observations influence the fitted acceleration through their sampling and weights. The gravity-to-height transfer example assumes an appropriate effective height has already been established; it does not derive one from an unweighted average.
What does a tiny residual tell me?
It supports agreement with the fitted curve for those observations under the stated weights. It does not independently validate the length scale, time scale, reference motion or every model assumption. A consistently mis-scaled trajectory can still be an excellent parabola.
Which check should I do first?
Use an exactly calculable trajectory, verify units and the factor of two, then shift the spatial and time origins. After those implementation tests, change the calibration scales and reference acceleration. Each test targets a different failure that a single repeated-fit statistic could miss.
Continue with the connected subjects
Connect acceleration to height and potential
Distinguish gravity-related potential differences from a geometric height
Carry shared model uncertainty through height comparisons
Preserve the reference and the information
Connect an instrument’s physical reference to other site marks
Retain shared nuisance uncertainty when combining estimates
Compare independent-error assumptions with time-correlated observations
Sources and scope
INRIM Gravity Laboratory: interferometric ballistic measurement and its metrological context
The numerical cases in this guide are original fictional teaching examples, not measurements from a real site. This is an educational explanation, not a field safety instruction, engineering approval or an official certification. For a real project, use qualified practitioners, current product documentation and site-specific evidence. Sources were checked on 1 November 2026.
Compare the physical quantities without merging them
A station’s support can change position, a falling body can reveal acceleration, clocks can constrain a potential difference, and a plumb line can establish direction. Read these as complementary measurements with different references, not interchangeable ways to report one height.
Follow the support and the acceleration
Thermal monument motion: connect temperature, support geometry and the moving point
Follow potential and direction
Optical clock comparisons: infer potential differences before converting to height
Deflection of the vertical: distinguish local plumb up from an ellipsoid normal