Experiment 7: Marble Collisions — Conservation of Momentum

Two marbles click together and, for an instant, exchange their motion. Hidden in that click is one of physics' great bookkeeping rules: in a collision, total momentum is conserved. This experiment tests the rule with ramps and slow-motion video, and uses the simulation — where speeds are exact — to verify it for equal masses, heavy-into-light and light-into-heavy collisions.

The physics behind the practical

Momentum is mass × velocity: p = mv, measured in kg·m/s, and it is a vector — direction matters, so we count motion to the right as positive and to the left as negative. During a collision the marbles exert equal and opposite forces on each other for the same brief time (Newton's third law again), so whatever momentum one loses, the other gains: total momentum before = total momentum after, provided outside forces (friction, gravity along a slope) have no time to matter.

The sim models elastic collisions, where kinetic energy is also conserved. The two conservation laws together fix the outcome completely. For equal masses with the target at rest, the only solution is a swap: the incoming marble stops dead and the target leaves at the full speed — the secret of the Newton's cradle. For unequal masses the elastic results are: vA′ = ((mA − mB)/(mA + mB)) × vA and vB′ = ((2mA)/(mA + mB)) × vA when B starts at rest. Real marble collisions are nearly elastic — you lose a little energy to the click of sound and a trace of heat — but momentum conservation does not depend on elasticity at all.

Variables and controls

In the virtual lab

Leave both masses at 50 g, set Marble A velocity 2.0 m/s and Marble B velocity 0, and press Collide! Compare the Momentum before and Momentum after readouts, and the A after / B after velocities: A stops, B leaves at 2.0 m/s. Now make B 100 g: A rebounds backwards (negative velocity) while B moves off more slowly. Make B 25 g and A keeps moving forwards after the hit. Finally give B a velocity of −1.0 m/s (rolling towards A) and predict before you collide. The Data Visualization chart's green "after" bar always matches the "before" total — the sim is exact, which makes it the perfect baseline for judging your noisier bench data. Log masses and all four velocities for at least three collisions.

Hands-on: ramp, marbles, slow motion

Build a ruler ramp propped on books, ending at a flat floor with a metre stick laid along the collision line. Rest marble B just beyond the ramp foot, release A from the marked height, and film from the side in slow motion. Estimate speeds from the video: pick two floor marks a known distance apart and divide by the time between frames as the marble passes them. Keep marbles off walkways between trials — they are a serious slipping hazard.

A worked investigation

In the sim (B at rest in each case, A's speed 2.0 m/s, so momentum before = 0.050 kg × 2.0 = 0.100 kg·m/s in the first row, and the same 0.100 target in all three):

Step 1 — Check the heavy-target case by formula. vA′ = ((50 − 100)/(50 + 100)) × 2.0 = (−1/3) × 2.0 = −0.67 m/s — the minus sign is the rebound. vB′ = (2×50/150) × 2.0 = 1.33 m/s.

Step 2 — Check momentum. Before: 0.100 kg·m/s. After: −0.033 + 0.133 = 0.100 kg·m/s. Conserved exactly, even though A now moves backwards — vector bookkeeping handles the sign automatically.

Step 3 — Compare with the bench. A real video measurement might give A arriving at 0.60 m/s and, after hitting an equal marble, A creeping at 0.05 m/s while B leaves at 0.52 m/s: total before 0.60m, after 0.57m (in units of the shared mass) — about 95% conserved, the missing 5% being friction during the approach and a slightly glancing contact. That gap is measurement reality, not a failed law.

Conclusion and evaluation

Across mass ratios, total momentum after matched total momentum before within measurement error, while speeds distributed exactly as the elastic-collision formulas predict. The dominant bench errors are speed estimation from video (frame timing and judging the marble's centre) and off-centre hits, which send marbles sideways and steal forward momentum from the 1-D accounting. Rolling friction before and after the click also bleeds a little speed. Improvements: a smooth, level track with guide rails for true head-on hits, and frame-by-frame video analysis with a scale in shot.

Safety

Collect marbles immediately after each trial and never leave them on the floor; keep the ramp stable and the filming position out of the rolling line.

Common mistakes

• Treating momentum as a scalar — a rebounding marble's momentum is negative; ignoring the sign "breaks" conservation that is actually fine.
• Glancing hits analysed as head-on — once a marble veers sideways, the one-dimensional sums no longer apply.
• Different release heights between trials, so A's arrival speed changes and comparisons wobble.
• Estimating speeds over different distances for A and B — use the same marks and the same method.
• Blaming the law for friction losses — conservation applies to the collision instant; friction acts before and after.

Exam Experience

Question 1 — calculation [4 marks]. In the simulation, marble A (50 g) moves at 2.0 m/s and hits stationary marble B (100 g) elastically. Calculate the total momentum before the collision and each marble's velocity afterwards, and verify that momentum is conserved.

Scoring guide:

• p before = 0.050 × 2.0 = 0.100 kg·m/s [1]
• vA′ = ((50 − 100)/150) × 2.0 = −0.67 m/s (rebound) [1]
• vB′ = ((2 × 50)/150) × 2.0 = 1.33 m/s [1]
• p after = 0.050 × (−0.67) + 0.100 × 1.33 = 0.100 kg·m/s = p before [1]

Question 2 — evaluation and design [4 marks]. A student's video analysis finds total momentum after a collision is consistently about 6% less than before. Give two physical reasons for the shortfall and two improvements to the method.

Scoring guide:

• Reason: rolling friction acts on both marbles before and after the collision, bleeding speed [1]
• Reason: slightly off-centre hits send momentum sideways, out of the measured line (or energy lost is a distractor — momentum loss needs an outside force/direction change) [1]
• Improvement: guide rails or a grooved track to guarantee head-on collisions on a level surface [1]
• Improvement: frame-by-frame video with a scale in shot and speeds measured over identical distances [1]

Multiple choice 1. A light marble hits a heavy stationary marble elastically. Afterwards the light marble:

• A. Stops dead
• B. Continues at its original speed
• C. Rebounds backwards
• D. Moves off with the heavy marble stuck to it

Answer: C. With mA < mB, ((mA − mB)/(mA + mB)) is negative, so A's velocity reverses — it bounces back while the heavy marble creeps forward.

Multiple choice 2. In a collision where the marbles stick together, kinetic energy falls noticeably. Total momentum:

• A. Falls by the same percentage
• B. Is still conserved (if outside forces are negligible during the impact)
• C. Doubles
• D. Becomes zero

Answer: B. Momentum conservation needs no elasticity — only negligible external impulse during the brief collision. Energy loss to heat and sound does not touch the momentum ledger.

Going deeper: energy in collisions — conserved or not?

Momentum survives every collision; kinetic energy is choosier, and comparing the two sharpens your reading of any impact. Take the sim's equal-mass case: KE before = ½ × 0.050 × 2.0² = 0.10 J, and after (A stopped, B at 2.0 m/s) KE = ½ × 0.050 × 2.0² = 0.10 J — perfectly elastic, nothing lost. Now the sticky case from the Try-it box (20 g at 1.0 m/s embedding in 10 g, leaving together at 0.67 m/s): KE before = ½ × 0.020 × 1.0² = 0.0100 J; KE after = ½ × 0.030 × 0.67² = 0.0067 J. A third of the kinetic energy vanished into deformation, heat and sound — while momentum, checked in the box, conserved exactly. Real collisions live on the spectrum between those extremes: marbles near the elastic end, car crashes engineered towards the sticky end.

That engineering is worth a sentence: crumple zones deliberately make collisions *less* elastic, stretching the impact over more time and converting kinetic energy into bent metal instead of into the passengers' momentum change arriving all at once. The momentum change is the same either way; the *rate* — the force — is what injures. One ledger (momentum) tells you what must happen; the other (energy) tells you what it costs. A complete collision analysis always audits both.

One more angle: when the target moves too

Head-on meetings check the vector rule hardest. In the sim, set A (50 g) at +2.0 m/s and B (50 g) at −1.0 m/s, rolling towards each other. Momentum before = 0.050 × 2.0 + 0.050 × (−1.0) = +0.050 kg·m/s. Elastic equal masses swap velocities: A leaves at −1.0 m/s, B at +2.0 m/s, so momentum after = 0.050 × (−1.0) + 0.050 × 2.0 = +0.050 kg·m/s — conserved, with both marbles rebounding from the meeting. The famous Newton's cradle is this swap chained: the end ball flies off because each internal collision hands the velocity along, ball to ball, almost losslessly.