SCIENCE · INVESTIGATE AND EXPLAIN

Test an energy target before taking a square root

An ideal spring stores energy kx²/2 when compressed by x from its relaxed length. The square matters: doubling compression multiplies stored energy by four. This model releases a virtual1 kg point from rest at height zero using a 40 N/m spring. After the spring relaxes it detaches, and a frictionless guide sends the point upward under g=10 m/s².

Learning outcome: Compare compressions0.5,0.8,1.0 m at a fixed target, then vary target at fixed compression. Retain the first inaccessible case and demonstrate algebraically that a common potential-energy offset cancels.

Starting recap

Choose the system, interval and reference frame before using a conservation statement. Keep vector momentum, scalar energy and transferred power distinct.

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Try

For x=0.5 m, stored energy is 5 J. At h=0.5 m the point reaches zero speed at its turning height. A target h=0.75 m is inaccessible and is not assigned negative kinetic energy.

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Predict

A negative computed kinetic energy at a proposed target means:

Your prediction and reason

Experiment

Compare compressions0.5,0.8,1.0 m at a fixed target, then vary target at fixed compression. Retain the first inaccessible case and demonstrate algebraically that a common potential-energy offset cancels.

Test whether spring energy can reach a selected height

Worked teaching model · synthetic values · Virtual 1 kg point is released from rest by a k=40 N/m ideal spring at height zero, then follows a frictionless guide upward under g=10. Spring is detached once relaxed. No residual rotation, impact loss or physical launch is included.

Initial spring compression: 0.8 m Target height above release level: 0.5 m

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Accessible post-release energy exchange

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: kinetic energy after spring relaxes (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Accessible post-release energy exchange

kinetic energy after spring relaxes

height (m)	kinetic energy (J)
0	12.8
0.032	12.48
0.064	12.16
0.096	11.84
0.128	11.52
0.16	11.2
0.192	10.88
0.224	10.56
0.256	10.24
0.288	9.92
0.32	9.6
0.352	9.28
0.384	8.96
0.416	8.64
0.448	8.32
0.48	8
0.512	7.68
0.544	7.36
0.576	7.04
0.608	6.72
0.64	6.4
0.672	6.08
0.704	5.76
0.736	5.44
0.768	5.12
0.8	4.8
0.832	4.48
0.864	4.16
0.896	3.84
0.928	3.52
0.96	3.2
0.992	2.88
1.024	2.56
1.056	2.24
1.088	1.92
1.12	1.6
1.152	1.28
1.184	0.96
1.216	0.64
1.248	0.32
1.28	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
Initial spring energy	12.8	J
Maximum accessible height	1.28	m
Selected height accessible	1	flag
Actual endpoint height	0.5	m
Endpoint gravitational energy	5	J
Endpoint kinetic energy	7.8	J
Endpoint speed	3.94968	m/s
Minimum compression for selected height	0.5	m

A target requiring more gravitational energy than the stored spring energy is inaccessible on this release.

Start with kx²/2, then compare mgh and mv²/2 after the spring is relaxed. The first zero of kinetic energy occurs at hmax. Adding the same constant reference to gravitational energy at both endpoints cancels from the balance; it cannot change the predicted speed or accessibility. A spring still attached at the endpoint would require retaining its remaining energy.

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

A target requiring more gravitational energy than the stored spring energy is inaccessible on this release.

Observation to attach to a capture

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Understand

Define the energy or momentum system

Build the algebraic account

Once the spring is relaxed, its initial energy is divided between kinetic energy and gravitational potential gain. Thus20x²=K+10h in joules. At a selected height h, calculate the required gravitational gain before taking a square root for speed. If10h exceeds 20x², the point cannot reach that height on this release; assigning an imaginary speed is a warning about the assumed endpoint.

Check an original packet

The first zero of kinetic energy occurs at hmax=2x². For x=0.8 m, initial energy is 12.8 J and maximum height is 1.28 m. At h=0.5 m, gravitational gain is 5 J, K is 7.8 J and speed is√15.6≈3.94968 m/s. For x=0.5 m, maximum height is 0.5 m; selecting a 1 m target must instead report the earlier turning height.

Test the domain and evidence

Gravitational potential may be shifted by any common additive constant. Adding100 J to its value at both endpoints cancels and cannot change speed or accessibility. By contrast, leaving a spring attached, adding friction or allowing rotation changes the physical energy terms, not just the reference. This ideal release is not an instruction to compress, launch or drop real equipment.

Worked original packet

For x=0.5 m, stored energy is 5 J. At h=0.5 m the point reaches zero speed at its turning height. A target h=0.75 m is inaccessible and is not assigned negative kinetic energy.

Bounded investigation

Changed-case numerical practice

Try the changed case before feedback

A 1 kg point receives8 J from an ideal spring and then rises without loss under g=10. At height 0.3 m, what kinetic energy remains, in joules?

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. All packets and diagrams are synthetic teaching evidence. No physical launches, collisions, spring releases, explosive processes or equipment-safety judgments are requested. Retained self-reports and public numerical feedback do not certify supervised laboratory practice or independent mastery.

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Why does an energy-reference shift differ from forgetting a remaining spring-energy term?

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