Conservation of momentum says that the total before equals the total after. But no measurement is exact. Your carts are timed to the nearest frame, your zones are marked to about a millimeter, and your scale reads to the nearest gram. So your before and after totals will never be identical, even if the law is exactly right.
That means you can’t test the law by asking “are the two numbers equal?” You test it by asking whether they agree within the uncertainty of your measurements. If the gap is smaller than the uncertainty, your data are consistent with the law. If the gap is bigger, something other than measurement noise is going on, and your job is to find it.
🔑A result that is “off” isn’t a failed experiment. A gap larger than your uncertainty is a clue that tells you something about the setup, and tracking it down is what this project is really about.
BMath
How Uncertain Is a Speed?
You find a speed from a distance and a time, v = d ÷ t, and each has an uncertainty. The zone is 0.50 m, good to about ±2 mm, which is ±0.4%. The time is the frame count divided by the frame rate, and you can only be sure to ±1 frame. For a product or quotient, the percent uncertainties add:
δv ÷ v = 1 ÷ frames + δd ÷ d
More frames means a smaller percent uncertainty. For a cart at 0.80 m/s crossing the zone in 0.625 s:
Frame rate
Frames counted
From ±1 frame
Total with the zone (±0.4%)
30 fps
19
5.3%
5.7%
60 fps
38
2.6%
3.0%
120 fps
75
1.3%
1.7%
240 fps
150
0.7%
1.1%
💡A slow-motion phone at 240 fps cuts the uncertainty from about 6% to about 1%. That is the difference between a lab that can detect a 5% momentum loss and one that can’t.
CMath
How Uncertain Is a Momentum?
Momentum is p = mv, a product, so the percent uncertainties add again: the mass percent plus the speed percent. For a total after a collision, find each cart’s δp and add them. Then the test is:
|p_after − p_before| ≤ δ_before + δ_after
Testing one collision
Given
Equal 0.250 kg carts, stuck together. Mass uncertainty ±0.001 kg
Before: 150 frames at 240 fps gives 0.80 m/s
After: 316 frames at 240 fps gives 0.38 m/s
Solve
Before: speed % = 1/150 + 0.4% = 1.07%, mass % = 0.4%, total = 1.47%
✓ The gap is more than twice the combined uncertainty, so the 5% loss is not measurement noise. Something systematic removed momentum.
⚠️The project’s Collision Verifier uses one safe speed uncertainty for every run, 1.5% by default. That is simpler and slightly more generous than the frame-by-frame numbers here, but the idea is the same.
DConcept
Random vs. Systematic Error
Not all errors behave alike. Telling them apart is the most useful skill in this lab:
Kind
Examples
What to do
Random
Reading a frame a little early or late; slightly different release each run; bumpers catching differently
Repeat the run and average. The scatter shrinks.
Systematic
Friction slowing the carts; a tilted track; a zone that is really 0.51 m; timing zones not adjacent to the line
Repeating doesn’t help. Find the cause, then fix the setup or correct for it.
The clue is the sign pattern. Random scatter makes some differences positive and some negative. A systematic error pushes them all the same way. If every setting shows less momentum after than before, friction is the first suspect, and your coast test is how you confirm it: a cart with no collision at all that still slows down.
EMath
Correcting for Friction
Friction slows the carts all the way along the track, so the speed in the before zone is not the speed at the collision line, and neither is the speed in the after zone. If the deceleration a is steady, the average speed across a zone equals the speed at the middle of the time in that zone. The line is half a zone-time away from that moment, so:
v_line, before = v_before − a t_before ÷ 2
v_line, after = v_after + a t_after ÷ 2
The cart is slower at the line than its average before it, and faster at the line than its average after it. You get a from the coast test, which gives the speeds in two zones in a row. It’s the change in speed divided by the time between the zone midpoints:
a = (v₁ − v₂) ÷ ((t₁ + t₂) ÷ 2)
Correcting the collision above
Given
Coast test: 0.80 m/s in zone 1 and 0.78 m/s in zone 2
The stuck-together collision above: p_before = 0.2000, p_after = 0.1900
✓ After correcting to the collision line, the gap (0.0029) is inside the uncertainty (0.0046). The 5% loss was friction.
⚠️The correction assumes steady friction, and it can’t be trusted for a cart that is almost stopped, because dividing by a tiny speed makes the zone time huge. If a cart barely moves after the collision, leave its speed as measured.
FExample
Uncertainty Explorer
Run a virtual collision. Start with 240 fps and a little friction, and watch the two error bars. Then drop the frame rate, raise the friction, and see what happens to the gap and to the correction.
A simulated lab: cart A hits cart B at rest and they stick. The tool “measures” both speeds from video with the frame rate you choose, including real friction, and then tests whether momentum was conserved. Try different frame rates and different amounts of friction.
Video frame rate:
Frames counted: before = 149 (±1.07% in speed), after = 313 (±0.72%) p before = 0.2013 ± 0.0030 · p after = 0.1917 ± 0.0018 gap = -0.0096 (-4.8%) · combined uncertainty = 0.0047
The gap is bigger than the uncertainty. Momentum came out low: a systematic loss.
After correcting for friction (a = 0.025 m/s²): p before = 0.1994, p after = 0.1998, gap = +0.0004
Corrected totals agree within uncertainty.
GConcept
Designing the Experiment Well
Good design removes systematic error before it happens and shrinks random error. Use this checklist:
Level the track and re-check it with the marble before each session.
Release cart A from exactly the same height every time.
Put the timing zones directly beside the collision line, so friction has little distance to act.
Use the highest frame rate you can, and record the raw frame counts.
Do three or more runs per setting, and keep them all. Drop a run only with a written reason.
Run a coast test, so you know what friction does with no collision at all.
💡About outliers: a run that looks wrong is information. Find out what happened, such as a cart that twisted or a bumper that missed. If you can name the cause, you may drop the run and say so. If you can’t, keep it.
HMath
Reporting a Verification
A good result states the numbers, the uncertainty, the comparison, and what the comparison means. For the example above:
Claim. Momentum was conserved in the equal-mass collision once friction is accounted for. Evidence. Raw totals were 0.2000 ± 0.0029 kg·m/s before and 0.1900 ± 0.0017 after, a 5% gap that is larger than the 0.0046 combined uncertainty. A coast test showed a friction deceleration of 0.032 m/s². Corrected to the collision line, the totals were 0.1975 and 0.2004, a gap of 0.0029, inside the uncertainty. Reasoning. The raw gap was always in the same direction, which points to a systematic cause. The coast test showed friction was large enough to explain it. Once removed, the two totals agree, so the data are consistent with conservation of momentum.
IWatch Out
Three Common Mix-Ups
⚠️“If the numbers don’t match exactly, the law failed.” They’re never going to match exactly. The question is whether the gap is bigger than your uncertainty. Even then, the more likely culprit is the setup, not the law.
⚠️“A perfect match proves a perfect experiment.” Be suspicious of one. With real measurements, agreement to the last digit usually means the uncertainty was hidden or the data were adjusted. An honest result has a small gap and a stated uncertainty.
⚠️“Uncertainty means I made a mistake.” It doesn’t. Uncertainty is the honest size of what your measurement can’t tell you. A mistake, like misreading a frame by ten, is a different thing, and the uncertainty doesn’t cover it.
JConcept
Quick Check
Four questions. They are not graded. Answer them to find out what still needs work.
Q1A cart crosses a 0.50 m zone in 150 frames of 240 fps video. What is its speed?
Q2You can only read the time to ±1 frame. What percent uncertainty does that give for a measurement that is 20 frames long?
Q3A group finds p_before = 0.200 ± 0.003 and p_after = 0.190 ± 0.003 kg·m/s. What should they conclude?
Q4Every one of a group’s five settings shows a momentum after that is 5–8% lower than before. What is the most likely explanation?
Score: 0 of 4
KConcept
Looking Ahead
You now know how to test a conservation law honestly, and how to say how well it holds. That’s the check you’ll need before you trust momentum conservation on a real crash.