C. Energy and Momentum · Momentum and Collisions

Investigation: Testing Conservation Laws in Collisions and Explosions

C2.7 C2.6 A1.1 A1.5 A1.6 A1.8 A1.10 A1.13About 75 minutes

Learning goals

I can plan an inquiry to test conservation of momentum and energy, with a hypothesis and variables.
I can calculate speeds from photogate or video data and find momentum and kinetic energy before and after.
I can compare results using percent differences and decide whether a difference is significant.
I can identify systematic and random errors and write a justified conclusion.

Words to know

photogate
A light beam and timer that measures how long an object blocks the beam.

air track
A track that blows air through small holes so gliders float with very little friction.

percent difference
|after − before| ÷ before × 100%; used to compare quantities that should be equal.

systematic error
An error that shifts every measurement in the same direction.

random error
Unpredictable scatter in measurements; reduced by repeating and averaging.

simulation
A computer model that calculates what happens using programmed physical laws.

Testing the laws yourself

You have used conservation of momentum dozens of times. But physics is an experimental science: a law is only as trustworthy as the tests it has passed. Your curriculum asks you to conduct laboratory inquiries or computer simulations with collisions and explosions in one and two dimensions, to test both conservation of momentum and conservation of energy. This lesson shows how a school lab does it, what real data look like, and how to analyse and judge the results. Then you will run a version at home.

Planning the inquiry

Question: In collisions and explosions between two gliders, is the total momentum the same before and after? Is the total kinetic energy the same?

Hypothesis (example): If two gliders collide on a nearly frictionless track, then their total momentum after will equal the total before (within a few percent) for every type of collision, because the only significant forces are internal. Total kinetic energy will be nearly unchanged with magnetic bumpers (nearly elastic), and about half lost when equal gliders stick with Velcro, because sticking converts kinetic energy into thermal energy and deformation.

Variables:

Independent: the type of interaction (Velcro, magnetic bumpers, spring-plunger explosion) and the mass ratio (add masses to a glider).
Dependent: the velocities of the gliders before and after.
Controlled: a level track, the same gliders and flags, and gates placed close to the collision point.

Equipment: an air track (or low-friction dynamics carts on a track), two gliders with flags, an electronic balance, and two photogates (light beams with timers). Many schools also use motion sensors or video analysis. When equipment is not available, the PhET Collision Lab simulation (free, from the University of Colorado Boulder) lets you run collisions in 1-D and 2-D and read velocities directly.

Safety: keep fingers clear of the spring plunger when it is cocked, do not lean on the track, and keep the air supply cord out of walkways.

A one-dimensional collision set-up. Each photogate measures how long a flag of known length blocks its light beam. Speed = flag length ÷ blocked time.

Sample results

The table below shows real-looking data for three trials. Glider A has mass 0.500 kg. Glider B has mass 0.500 kg (trial 1) or 0.750 kg (trials 2 and 3). Right is positive.

Trial	v_A (m/s)	v_A′ (m/s)	v_B′ (m/s)	p before (kg·m/s)	p after (kg·m/s)	Ek before (J)	Ek after (J)
1: Velcro, equal masses	0.80	0.39	0.39	0.400	0.390	0.160	0.076
2: magnets, B = 0.750 kg	0.80	−0.15	0.62	0.400	0.390	0.160	0.150
3: explosion, B = 0.750 kg	0	−0.42	0.27	0	−0.0075	0	0.071

Sample glider data. Each velocity is the average of three runs.

Worked example

Example 1: Analysing trials 1 and 2

Use the table to test conservation of momentum and to classify the collisions in trials 1 and 2.

Trial 1 momentum: before (0.500)(0.80) = 0.400; after (1.000)(0.39) = 0.390 kg·m/s. Percent difference = |0.390 − 0.400| ÷ 0.400 × 100% = 2.5%.
Trial 1 kinetic energy: before ½(0.500)(0.80)² = 0.160 J; after ½(1.000)(0.39)² ≈ 0.076 J, which is 47.5% of the original. Theory for a perfectly inelastic collision of equal masses: 50%.
Trial 2 momentum: after (0.500)(−0.15) + (0.750)(0.62) = −0.075 + 0.465 = 0.390 kg·m/s: again 2.5% below.
Trial 2 kinetic energy: ½(0.500)(0.15)² + ½(0.750)(0.62)² ≈ 0.0056 + 0.1442 = 0.150 J, which is 93.6% of the original.
Theory for trial 2 (elastic): v_A′ = [(0.500 − 0.750) ÷ 1.250](0.80) = −0.16 m/s and v_B′ = [1.000 ÷ 1.250](0.80) = 0.64 m/s. The measured −0.15 and 0.62 m/s are close.

Momentum agrees within 2.5% in both trials, supporting conservation of momentum. Trial 1 kept about 48% of its kinetic energy (perfectly inelastic, as expected); trial 2 kept about 94% (nearly elastic).

Trials 1 and 2 as bar charts. Momentum (p bars) is almost the same before and after in both trials. Kinetic energy (Ek bars) is roughly halved in the Velcro collision but nearly unchanged with magnets. (Momentum and energy bars use different scales.)

Worked example

Example 2: Analysing the explosion

In trial 3, the gliders start at rest and a spring plunger pushes them apart. Test conservation of momentum, and find the energy released.

Before: p = 0.
After: (0.500)(−0.42) + (0.750)(0.27) = −0.210 + 0.2025 = −0.0075 kg·m/s.
A percent difference from zero cannot be calculated, so compare with the size of each glider's momentum: 0.0075 ÷ 0.21 ≈ 3.6%.
Kinetic energy after: ½(0.500)(0.42)² + ½(0.750)(0.27)² ≈ 0.044 + 0.027 = 0.071 J, which came from the spring's elastic potential energy.

The total momentum after (−0.0075 kg·m/s) is only about 4% of either glider's momentum, so it is zero within experimental error. The spring released about 0.071 J.

A two-dimensional trial

On an air table (or with pucks on a smooth surface filmed from above), a 0.10 kg puck moving at 0.60 m/s [E] strikes an identical puck at rest. Measurements from the video: puck A leaves at 0.45 m/s [E 40° N] and puck B at 0.37 m/s [E 52° S].

x-momentum after: (0.10)(0.45 cos 40° + 0.37 cos 52°) ≈ (0.10)(0.345 + 0.228) = 0.0573 kg·m/s, compared with 0.060 before: 4.6% low.
y-momentum after: (0.10)(0.45 sin 40° − 0.37 sin 52°) ≈ (0.10)(0.289 − 0.292) ≈ −0.0002 kg·m/s: zero within error.
Kinetic energy: before ½(0.10)(0.60)² = 0.0180 J; after ½(0.10)(0.45² + 0.37²) ≈ 0.0170 J: 94% kept.
The angle between the paths is 40° + 52° = 92°, close to the 90° expected for an elastic collision of equal masses.

Judging the results: sources of error

Systematic errors push results the same way every time. Friction (air drag on gliders, track friction on carts) slows the gliders between the collision and the gate, so the measured momentum after is a little low in every trial, exactly the pattern in the data. Fix: place gates close to the collision, or measure the slowing and correct for it.
Random errors scatter results: timing a flag to the nearest millisecond, slightly different pushes, reading a video frame. Fix: repeat trials and average.
Measurement uncertainty: a percent difference smaller than the combined uncertainty of your measurements is not significant. If speeds are uncertain by about ±2% and masses by ±0.5%, momentum is uncertain by about ±2.5%, so a 2.5% difference is consistent with conservation.

Significant digits: report results to the precision of your weakest measurement (here, usually 2 significant digits for speeds).

Worked example

Example 3: Writing the conclusion

Write a conclusion for the glider investigation.

Answer the question: total momentum after each interaction was within about 2.5% to 4.6% of the total before, including the explosion and the 2-D collision, where y-momentum stayed zero within error.
Compare with the hypothesis: this supports the hypothesis that momentum is conserved in all three kinds of interaction.
Energy: kinetic energy was nearly conserved with magnetic bumpers (94%) and in the 2-D puck collision (94%), and about half was lost when equal gliders stuck (48%, theory 50%). The explosion gained 0.071 J, from the spring.
Errors: momentum after was always slightly low, consistent with a systematic error from friction acting between the collision and the second gate.
Improvement: move the gates closer to the collision point, and measure each glider's deceleration on its own to correct the speeds.

A strong conclusion answers the question with numbers, links back to the hypothesis, explains the pattern of errors and suggests a specific improvement.

Just for fun

An air track floats gliders on a thin cushion of air blown out through hundreds of tiny holes. It is basically an air hockey table that has been stretched into one dimension, which may be the only time physics made a game less fun on purpose.

Hands-on activity

Kitchen-floor collision lab

You need: two identical toy cars (or two identical full soup cans laid on their sides), modelling clay or poster putty, a kitchen scale, a metre stick, masking tape, and a phone that records slow-motion video (or a computer with the free PhET Collision Lab simulation).

Weigh each car. Tape the metre stick along a smooth floor as a scale and set the phone on a stack of books to film from the side, close to floor level.
Elastic-ish collision: place car B at rest beside the scale. Push car A so it rolls into B. Film it.
Sticking collision: put a small blob of clay on the front of A so the cars stick, and repeat.
Explosion: tape a strip of thin card into a loop between the two cars, squeeze them together and let go so the card pushes them apart. Film it.
Step through each video frame by frame. Speed = distance moved ÷ (number of frames ÷ frames per second), measured over a short distance just before and just after the collision.
Calculate total momentum and total kinetic energy before and after, with percent differences.

With the simulation instead: set up the same three cases, plus a 2-D collision with "elasticity" set to 100% and then to 0%, and record the values it shows.

What to look for: momentum before and after should agree to within about 10% with toy cars (friction and wobbly wheels add error), and the explosion's total should be close to zero. Kinetic energy should drop noticeably when the cars stick. In the simulation, momentum is conserved exactly; kinetic energy is conserved only at 100% elasticity.

Your turn

A glider's 0.100 m flag blocks a photogate for 0.125 s. Find the glider's speed.

The total momentum before a collision was 0.420 kg·m/s and after was 0.401 kg·m/s. Find the percent difference.

A 0.40 kg cart at 0.60 m/s hits an identical cart at rest; they stick and move at 0.28 m/s. Test conservation of momentum and find the fraction of kinetic energy kept. What does theory predict?

In an inquiry testing conservation of momentum for different mass ratios, name the independent, dependent and controlled variables.

In an explosion trial, a 0.25 kg cart moves at 0.36 m/s [left] and a 0.50 kg cart at 0.17 m/s [right]. Is momentum conserved within experimental error?

Why should each trial be repeated several times?

In every trial, the measured momentum after a collision is slightly less than before. Suggest a cause and a fix.

More practice: warm-up

What does a photogate measure?

Write the formula for percent difference between a value before and after.

What should you compare to test whether a collision is elastic?

Is friction on a track a random or a systematic error?

Why use an air track?

More practice: on your own

Two 0.20 kg pucks: A moves at 0.80 m/s [E] and hits B at rest. Afterward A moves at 0.50 m/s [E 50° N] and B at 0.62 m/s [E 38° S]. Test both conservation laws.

Write a hypothesis for an inquiry comparing collisions of carts with Velcro and with magnetic bumpers.

A student's Velcro trial shows more kinetic energy after the collision than before. What has probably gone wrong?

Video at 240 frames per second shows a cart crossing 0.20 m in 60 frames. Find its speed and the percentage uncertainty if the frame count could be off by ±1 frame.

Give one advantage and one disadvantage of a computer simulation compared with a real lab.

Challenge

Mass is measured to ±0.5% and speed to ±2%. Estimate the uncertainty in a momentum, and decide whether a 2.5% difference between momentum before and after is significant.

Design a test of the 90° rule for equal masses, using two identical coins (or pucks) and a phone camera mounted overhead.

A cart leaves a collision at 0.40 m/s. Friction slows it at 0.050 m/s², and the photogate is 0.30 m beyond the collision point. What speed does the gate record, and by what percentage does this understate the momentum?

Check your understanding

Quiz

Why is an air track used to test conservation of momentum?

A 0.050 m flag blocks a photogate for 0.020 s. The glider's speed is:

Two equal gliders stick together. The data show momentum unchanged within 2% and kinetic energy reduced to 49%. The best conclusion is:

Total momentum is 0.400 kg·m/s before a collision and 0.390 kg·m/s after. The percent difference is:

Which is an example of a systematic error in a glider experiment?

In an explosion trial starting from rest, the evidence that momentum is conserved is that:

For parents and teachers

Big idea (C2.7, A1): conservation laws are tested by experiment. Your teen measured (or analysed) collisions and explosions, compared momentum and kinetic energy before and after with percent differences, and judged the results using sources of error and uncertainty.

How to help: Help set up the phone and toy cars for the kitchen-floor lab, and ask your teen to explain one systematic error and one random error in their own results. If no suitable toys are available, the free PhET Collision Lab simulation runs in a web browser.

Common difficulties:

Percent difference from zero (explosions): compare with the size of each object's momentum instead.
Over-claiming. "Proves" is too strong; data support a law within experimental error.
Vague error analysis. Encourage specific causes with a direction ("friction made speeds after the collision lower") and specific fixes.
Significant digits. Results should not have more digits than the measurements justify.