SCIENCE · INVESTIGATE AND EXPLAIN

Test numerical update order against an exact trajectory

A plausible plotted curve is not evidence that a time integrator is accurate.

Learning outcome: Compare all three update rules at one N, then double N. Preserve exact and computed endpoints, error sign and the distinction between trajectory error and an impact model.

Starting recap

Read derivatives as local rates and integrals as signed accumulation. Review power and chain rules, vector components, units and initial conditions where used.

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Try

With N=20, dt=.05 s, forward and velocity-first endpoint errors are+.25 and−.25 m. Doubling N halves these endpoint errors.

Open the experiment

Predict

At N=10, forward Euler endpoint height error is:

Your prediction and reason

Experiment

Compare all three update rules at one N, then double N. Preserve exact and computed endpoints, error sign and the distinction between trajectory error and an impact model.

Compare update order against an exact projectile

Worked teaching model · synthetic values · Virtual trajectory begins x0=0,y0=1 m with velocity(3,4) m/s and constant acceleration(0,−10) m/s². Exact first ground crossing is1 s. Integrators are compared algebraically to that endpoint; negative numerical height is an error, not simulated penetration or an impact response.

Equal steps over1 s: 10 steps Update:0 forward Euler,1 velocity-first,2 midpoint: 0 method

Reset keeps your prediction, captures and practice history.

Time-stepped path against exact pre-impact motion

Spatial axes use the same pixels per physical unit, preserving geometry. 1: computed algebraic steps to1 s (solid).2: exact path to ground (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Time-stepped path against exact pre-impact motion

computed algebraic steps to1 s

east position (m)	height (m)
0	1
0.3	1.4
0.6	1.7
0.9	1.9
1.2	2
1.5	2
1.8	1.9
2.1	1.7
2.4	1.4
2.7	1
3	0.5

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

exact path to ground

east position (m)	height (m)
0	1
0.0375	1.049219
0.075	1.096875
0.1125	1.142969
0.15	1.1875
0.1875	1.230469
0.225	1.271875
0.2625	1.311719
0.3	1.35
0.3375	1.386719
0.375	1.421875
0.4125	1.455469
0.45	1.4875
0.4875	1.517969
0.525	1.546875
0.5625	1.574219
0.6	1.6
0.6375	1.624219
0.675	1.646875
0.7125	1.667969
0.75	1.6875
0.7875	1.705469
0.825	1.721875
0.8625	1.736719
0.9	1.75
0.9375	1.761719
0.975	1.771875
1.0125	1.780469
1.05	1.7875
1.0875	1.792969
1.125	1.796875
1.1625	1.799219
1.2	1.8
1.2375	1.799219
1.275	1.796875
1.3125	1.792969
1.35	1.7875
1.3875	1.780469
1.425	1.771875
1.4625	1.761719
1.5	1.75
1.5375	1.736719
1.575	1.721875
1.6125	1.705469
1.65	1.6875
1.6875	1.667969
1.725	1.646875
1.7625	1.624219
1.8	1.6
1.8375	1.574219
1.875	1.546875
1.9125	1.517969
1.95	1.4875
1.9875	1.455469
2.025	1.421875
2.0625	1.386719
2.1	1.35
2.1375	1.311719
2.175	1.271875
2.2125	1.230469
2.25	1.1875
2.2875	1.142969
2.325	1.096875
2.3625	1.049219
2.4	1
2.4375	0.9492188
2.475	0.896875
2.5125	0.8429687
2.55	0.7875
2.5875	0.7304688
2.625	0.671875
2.6625	0.6117188
2.7	0.55
2.7375	0.4867188
2.775	0.421875
2.8125	0.3554688
2.85	0.2875
2.8875	0.2179687
2.925	0.146875
2.9625	0.07421875
3	0

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Step size	0.1	s
Computed endpoint height	0.5	m
Exact endpoint height	0	m
Signed endpoint height error	0.5	m
Computed final vertical velocity	-6	m/s
Exact first ground-crossing time	1	s

Identical acceleration and step sizes do not make different update orders identical.

Forward Euler endpoint error is+5dt, velocity-first is−5dt and the midpoint-velocity update is exact for this constant acceleration apart from floating rounding. Negative numerical height diagnoses discretization error, not a modeled impact or physical penetration. All methods give the same final velocity here because integrating a constant acceleration is exact.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Identical acceleration and step sizes do not make different update orders identical.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Test numerical update order against an exact trajectory

A plausible plotted curve is not evidence that a time integrator is accurate.

The original virtual trajectory starts at(0,1) m with velocity(3,4) m/s and acceleration(0,−10) m/s². Exact motion is x=3t and y=1+4t−5t². The first positive ground root is t=1 s, with vertical velocity−6 m/s. This known endpoint lets us test numerical error without confusing it with an unknown physical parameter.

Forward Euler updates position using the old velocity and then updates velocity. With N equal intervals over1 s, dt=1/N, it overestimates final height by5dt. A velocity-first update instead uses the new velocity for displacement and underestimates height by5dt. Both integrate the constant acceleration exactly for velocity, so a correct final velocity does not establish a correct position.

A midpoint-velocity position update uses v_old+a dt/2 during each interval. For this constant acceleration it reproduces the exact quadratic positions apart from floating rounding. This special exactness is not a claim about arbitrary position-dependent or time-dependent acceleration, for which method order and stability need another analysis.

At N=10, forward Euler reports +.5 m and velocity-first−.5 m at the exact impact time; midpoint gives zero. The negative computed height is a numerical discrepancy, not a simulated penetration or an impact response. Compare the algebraic trajectories only through the declared1 s benchmark; a real impact model would require event detection and new physics.

Changed-case check

Predict before reading the comparison. With N=20, dt=.05 s, forward and velocity-first endpoint errors are+.25 and−.25 m. Doubling N halves these endpoint errors.

Retained investigation

Compare all three update rules at one N, then double N. Preserve exact and computed endpoints, error sign and the distinction between trajectory error and an impact model. Why can final velocity pass while final position fails?

Source and evidence boundary

Conceptual reference checked30 October2026: OpenStax University Physics Volume1. All prose, examples and models are original. These are bounded supplied mathematical trajectories or synthetic observations, not observed physical experiments. No launch, impact or apparatus procedure is requested. Record assumptions, prior predictions, errors and independent checks; guided success is not independently demonstrated proficiency.

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Public practice checks a specific claim. It does not establish independent mastery.

Why can final velocity pass while final position fails?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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Previous: Differentiate the basis vectors of circular motion Next: Compare integration rules with an exact displacement

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