This article was originally published on LinkedIn, October 5, 2025: https://www.linkedin.com/pulse/grows-doubling-halving-tom-regan-uuwre
Introduction
The 2028 NAEP Science Assessment Framework contains this description of kinetic energy:
P8.12: The energy of motion of particles or waves is called kinetic energy; for massive objects it is proportional to the mass of the moving object and grows with the square of its speed.[1]
This language originated with the NRC’s K-12 Framework[2] in 2012. Thirteen years later, I still find it fascinating. Why does the K-12 Framework employ straightforward, unambiguous language for the mass dependence, but the enigmatic “grows with” construction for the speed dependence? And what is the functional dependence of kinetic energy on speed, according to this language?
What Does It Mean?
Let’s consider the second question first. I will describe an interpretation of “grows with the square of its speed” that seems reasonable based solely on the language. However, this interpretation will lead to incorrect physics.
Suppose an object’s initial speed and kinetic energy are both 0. Then the speed increases to 1; the square of 1 is 1. According to this interpretation, 1 is the growth in the kinetic energy, so the new kinetic energy is 0 + 1 = 1.
The speed increases to 2, 2 squared is 4, so 4 is the growth in the kinetic energy. The new kinetic energy is 1 + 4 = 5.
The speed increases to 3, 3 squared is 9, and the new kinetic energy is 5 + 9 = 14.
Continuing in this manner, we obtain these data, where the first value is the speed and the second is the kinetic energy: (0,0), (1,1), (2,5), (3,14), (4,30), (5,55), (6,91), (7,140), (8,204), (9,285), (10,385). The second-order polynomial fit is , . The fit looks good. The points are close to the plot of the curve, though not exactly on it.
However, the third-order fit, , is even better. , the residuals are zero, the data lie exactly on the regression curve. A perfect fit! Apparently, the dependence of kinetic energy on speed is third order: .
Though this may be a reasonable interpretation of “grows with,” the resulting physics is incorrect. The equation for kinetic energy () is . The “grows with” language should be interpreted like this: Suppose that the starting speed is 1 and the starting kinetic energy is 100. As the speed increases to 2, 3, and 4, the kinetic energies follow the pattern of the squares: 100, 400, 900, and 1600.
The point is that the language “grows with the square of its speed” is ambiguous. It can be interpreted in ways that are reasonable on the grounds of language alone, but that lead to incorrect physics.
Why Was It Written This Way?
Why was this ambiguous language chosen? A parallel construction would have been much clearer:
An object’s kinetic energy is proportional to the first power of its mass and the second power of its speed.
Or, for maximum clarity, simply stating the equation is best: .
I believe the answer is that the K-12 Framework writer was indicating what should and should not be assessed.
The Evidence Statements for Performance Expectation (PE) MS-PS3-1[3] of the Next Generation Science Standards[4] (NGSS) provide useful context for considering the wording in the K-12 Framework. The PE is:
MS-PS3-1. Construct and interpret graphical displays of data to describe the relationships of kinetic energy to the mass of an object and to the speed of an object.
The Evidence Statement begins:
1.a.i,ii,iii: Students use graphical displays to organize the…mass…speed…[and] kinetic energy of the object.
Notice that students are organizing, but not necessarily calculating, kinetic energy. The intention may be that the graphing task provides the values of kinetic energy to the student. This suggests a plausible reason why the K-12 Framework language and the MS-PS3-1 PE and Evidence Statement do not include the equation for kinetic energy. They do not want to imply that calculating kinetic energy using the equation will be assessed.[5]
Next, a speed-related part of the Evidence Statement reads:
3.a.ii.1: Using the analyzed data, students describe the relationship between kinetic energy and speed as a nonlinear (square) proportional relationship (KE α v2) in which the kinetic energy quadruples as the speed of the object doubles.[6]
If the speed doubles, the kinetic energy “grows with the square of [the] speed,” meaning that it quadruples. I bet that the K-12 Framework writer wanted students to demonstrate this dynamic, growth- or change-based reasoning.
Change Is Not Always Good
If dynamic reasoning is desirable, why didn’t the K-12 Framework use similar language for the mass dependence, something like “the kinetic energy grows with the mass” or “grows linearly with the mass”?
I have a guess. The mass dependence part of the Evidence Statement reads:
3.a.i.1,2: Using the analyzed data, students describe the relationship between kinetic energy and mass as a linear proportional relationship (KE α m) in which the kinetic energy doubles (halves) as the mass of the object doubles (halves).
It is surprisingly difficult to conceptualize an object’s mass changing. Suppose a stone’s mass is 1 kilogram (kg), and its speed is 10 meters per second (m/s). Its kinetic energy is 50 joules (J). Now, suppose its mass doubles—but how does the mass of a stone double? Stones do not grow. Maybe a stone is not a good example.
Let’s try a truck instead. Suppose the truck’s mass is 2,000 kg[7] and it rolls toward a gravel-filled hopper at a speed of 2 m/s. When empty, the truck’s kinetic energy is 4,000 J. Then the hopper dumps 2,000 kg[8] of gravel into the truck’s bed, and the truck’s mass becomes 4,000 kg. However, this scenario presents issues. First, because it involves conservation of momentum, it is best addressed when studying momentum. Second, if the gravel has no horizontal velocity when dumped, momentum analysis reveals that the resulting speed of the truck + gravel is 1 m/s, and the resulting kinetic energy is 2,000 J. The kinetic energy of the truck hasn’t doubled; in fact, it has decreased.
Let’s try the truck again, this time halving rather than doubling the mass. Our 2,000-kg truck tows a 2,000-kg[9] car at a speed of 30 m/s. The total mass is 4,000 kg, and the kinetic energy is 1,800,000 J. But the truck hits a bump, the tow hitch fails, and the car rolls free, for the moment still traveling at 30 m/s. We might be tempted to say that the mass of the truck + car is half of what it was. However, this doesn’t make sense because the truck + car no longer exists. In place of the original composite object there are now two different objects, each with half the mass. Each carries half of the original kinetic energy, so linear proportionality holds; that’s not the issue. The problem is that the context of an object gaining or losing mass doesn’t make sense. In general, when an object’s mass changes, it becomes a different object (or two).[10]
Fortunately, there is a different, perfectly adequate way to assess the dependence of kinetic energy on mass: simply compare two objects. For example, if Trucks A and B are traveling at the same speed, and Truck A’s mass is twice that of Truck B, how does Truck A’s kinetic energy compare to Truck B’s? I suspect, however, that some may view static comparisons as valuable than dynamic reasoning, even when the underlying science is the same.
For completeness, I believe that saying that a truck’s speed increases is fine. We know how this happens—the truck burns a little fuel. The mass of the fuel consumed is negligible compared to the truck’s mass, so it is meaningful to say that it is still the same truck. In contrast, for a rocket, the fuel is the bulk of its mass. Exercises in proportionality should avoid using rockets as examples.
Summing Up and Final Thoughts
I believe the NRC K-12 Framework description of kinetic energy was written to discourage memorization and calculation, and instead to encourage dynamic, growth- or change-based reasoning. I would like to think that the mass dependence was described differently from the speed dependence because the writer realized that a dynamic context is not a good fit for an object’s mass.
Having proposed a distinction between static and dynamic comparisons, I searched for an existing knowledge base and was surprised to find nothing. None of the K-12 Framework/NGSS Science and Engineering Practices seem to address this distinction, and a cursory Internet search didn’t turn up anything. Even Copilot didn’t quite get the point. I hope that knowledgeable readers can enlighten me.
It is unfortunate that the 2028 NAEP Framework uses the ambiguous “grows with” language. However valuable the associated reasoning may be, the language is simply not clear. For future documents and assessments, the NGSS Evidence Statement regarding speed provides a good model. For the mass dependence of kinetic energy, a static comparison of two objects is preferable.
[1] 2028 NAEP Science Assessment Framework, page 21. https://www.nagb.gov/content/dam/nagb/en/documents/publications/frameworks/science/2028-naep-science-framework.pdf
[2] National Research Council 2012. A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas. Washington, DC: The National Academies Press. https://doi.org/10.17226/13165. Page 123. https://nap.nationalacademies.org/catalog/13165/a-framework-for-k-12-science-education-practices-crosscutting-concepts
[4] NGSS Lead States. 2013. Next Generation Science Standards: For States, By States. Washington, DC: The National Academies Press. https://www.nextgenscience.org/
[5] In fact, the equation does not appear anywhere in the K-12 Framework or the NGSS.
[6] The exponent, the 2 in v2, is not superscripted. That’s not my typo; that’s how it appears, incorrectly, in the NGSS. This is very disappointing. The whole point of the PE is the relationship of kinetic energy to mass and speed. The exponent is the relationship. It is the most important typography on the page, and it is botched.
[7] Considering the Ford F-150 as an example, curb weights vary depending on configuration, but the lightest model is 1,998 kg. https://www.ford.com/trucks/f150/specs/
[8] This value actually is too high; maximum payload for the Ford F-150 is around 1,000 kg.
[9] This value is reasonable; the Ford F-150 can tow at minimum 3,360 kg, and up to twice that depending on configuration.
[10] For these reasons, graphs having mass on the x-axis deserve extra scrutiny to ensure that the context makes sense.