This article was originally published on LinkedIn, August 22, 2025: https://www.linkedin.com/pulse/conserved-tom-regan-vlshe
The Next Generation Science Standards[1] (NGSS) Performance Expectation HS-PS2-2[2], “Use mathematical representations to support the claim that the total momentum[3] of a system of objects is conserved when there is no net force on the system”, is phrased incorrectly. There are two possible fixes.
If the goal is simply to describe the linear momentum of an isolated system, the word “constant” should replace “conserved”. It’s true that if there is no net force on a system, then its linear momentum is constant.
A more ambitious goal is to highlight a very important property of linear momentum: it is one of the few quantities that is conserved. This means:
The change in a system’s linear momentum equals the net flow of linear momentum into the system.
If this is the goal, then the statement should be: “Linear momentum is a conserved quantity”, or “Linear momentum is conserved”.
Conservation may be easier to appreciate when contrasted with non-conservation. A quantity that is not conserved is volume. Consider an old-style thermometer with a column of mercury or red-tinted alcohol. After the thermometer sits in a refrigerator for a while, the column reaches the 4°C mark. Now remove the thermometer from the refrigerator and set it on the counter. The column rises to the 22°C mark. The column’s volume increases, though no liquid enters it.
In the language of our definition of conservation, the column of liquid is the system. The change of the system’s volume, which is some positive number, does not equal the net flow of volume into the system, which is zero. Thus, volume is not a conserved quantity.
This brings me to a pet peeve: algebra problems that (incorrectly) assume that volume is conserved when different liquids are mixed. For example:
“What volumes, in liters, of 20% and 80% ethanol solutions must be mixed to obtain 2 liters of 60% ethanol solution?”
Students are expected to let x be the volume of the 20% solution and y be the volume of the 80% solution, and write a system of equations:
(1) Ethanol volume: 0.2 x + 0.8 y = 0.6 * 2
(2) Total volume: x + y = 2.
The solution 😊 to this system is x = 2/3 L, y = 4/3 L.
Unfortunately, volume is not conserved, so equation (2) is invalid. Adding 2/3 L of one liquid to 4/3 L of another does not generally yield 2 L. The total volume typically will be slightly less. This is not magic; it’s just a consequence of the molecules arranging themselves a little differently when their neighboring molecules are different.
Volume isn’t conserved, but mass is, and liquid-mixing problems can be solved by keeping track of the masses of the components using the densities of the liquids. Such an analysis[4] shows that mixing 2/3 L of 20% ethanol solution with 4/3 L of 80% solution yields a final volume of about 1.98 liters, roughly 1% less than the arithmetic sum of the initial volumes.
I hope that the math education community will reconsider using problems, like liquid-mixing problems, that assume volume conservation. A 1% discrepancy may seem small, but it represents the difference between conservation and non-conservation, and that’s huge.
[1]. NGSS Lead States. 2013. Next Generation Science Standards: For States, By States. Washington, DC: The National Academies Press. https://www.nextgenscience.org/
[2]. HS-PS2-2 Motion and Stability: Forces and Interactions | Next Generation Science Standards
[3]. The reference to “net force” indicates that this PE concerns linear momentum. If it concerned angular momentum, then the reference would be to net torque. This distinction matters not because angular momentum isn’t conserved—it is—but because the NGSS do not include assessment of angular momentum.
[4]. Densities from https://www.handymath.com/calcshtml/etohtable2.html.
- Concentrations are volume EtOH per volume solution
- Water at 20°C: 0.99823 g/mL
- 20% ethanol solution: 0.97359 g/mL
- 80% ethanol solution: 0.85929 g/mL
- Ethanol at 20°C: 0.78934 g/mL
- Volume of 20% EtOH solution 0.6667 L; 80% EtOH solution 1.3333 L
- Volume of EtOH in 20% solution 0.13334 L; 80% solution 1.06664 L
- Mass EtOH in 20% solution 0.10525 kg; 80% solution 0.84194 kg
- Mass 20% solution 0.64909 kg; 80% solution 1.14569 kg
- Mass water in 20% solution 0.54384 kg; 80% solution 0.30375 kg
- Total mass of mixture 1.79478 kg
- Mass EtOH in mixture 0.94719 kg; water 0.84759 kg
- Mass % EtOH of mixture 52.77%
- Density of mixture 0.90763 g/mL (from table)
- Volume of mixture 1.97744 L
- Arithmetic sum of initial volumes 2.0000 L
- Excess volume -0.02256 L (-1.14%)
- Volume percent ethanol 60.678%. (Using the density to interpolate on the % Volume Ethanol table).