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Designing Momentum
Conservation Experiments

Learn to design rigorous experiments that test the law of conservation of momentum: \( \sum \vec{p}_i = \sum \vec{p}_f \).

High School Physics

What Is Momentum?

Momentum is the product of an object's mass and velocity. It is a vector quantity: it has both magnitude and direction.

Scalar Component
Mass \( m \): always positive, measured in kilograms
Vector Component
Velocity \( \vec{v} \): has magnitude and direction (m/s)

The Conservation Law

In a closed, isolated system (no external net force), the total momentum before any interaction equals the total momentum after.

This law holds for all types of collisions: elastic, inelastic, and perfectly inelastic, as well as explosions where objects separate.

Why Experiment Design Matters

Simply knowing the formula isn't enough. A well-designed experiment lets you test whether momentum is truly conserved in your system and quantify sources of error.

Hypothesis
Formulate a testable prediction: total momentum before = total momentum after
Variables
Identify what you measure (velocities), control (surface friction), and vary (masses)
Evidence
Collect quantitative data and compare \( \sum p_i \) to \( \sum p_f \) with error analysis

Momentum Before = Momentum After

Before Collision \(m_1\) \(v_{1i}\) \(m_2\) \(v_{2i}=0\) → COLLISION After Collision \(m_1\) \(v_{1f}\) \(m_2\) \(v_{2f}\)
\( m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \)

Elastic vs Inelastic Collisions

Elastic Inelastic Perfectly Inelastic
Momentum Conserved Conserved Conserved
Kinetic Energy Conserved Not conserved Not conserved
Objects After Bounce apart Separate (deformed) Stick together
Example Billiard balls Car crash (cars separate) Clay balls colliding

Identifying Key Variables

Independent
What you deliberately change: cart masses, initial velocities, collision type
Dependent
What you measure: final velocities of each object after the collision
Controlled
What you keep constant: surface friction, track angle, measurement tools

How to Design the Experiment

1
State your hypothesis: "If no external forces act, then \( \sum p_i = \sum p_f \) within measurement uncertainty."
2
Select equipment: Two carts on a low-friction track, mass balance, photogates or motion sensors to measure velocities.
3
Measure initial conditions: Record masses \( m_1, m_2 \) and initial velocities \( v_{1i}, v_{2i} \). Compute \( p_{\text{total,i}} \).
4
Perform the collision and record final velocities \( v_{1f}, v_{2f} \). Compute \( p_{\text{total,f}} \).
5
Repeat multiple trials (at least 5) and calculate percent difference: \( \%\text{diff} = \frac{|p_f - p_i|}{p_i} \times 100\% \).
6
Analyze: If \( \%\text{diff} \) is small (typically < 5%), momentum is conserved within experimental error.

Typical Lab Setup

Low-Friction Track Cart 1 \(v_{1i}\) Cart 2 at rest Gate A Gate B Gate C
Measures
\(v_{1i}\)
 Collision Zone
Measures
\(v_{1f}, v_{2f}\)
 Mass Balance Measure \(m_1, m_2\)

Example 1: Perfectly Inelastic

Cart 1 (\( m_1 = 0.50 \) kg) moves at \( v_{1i} = 2.0 \) m/s toward stationary Cart 2 (\( m_2 = 0.50 \) kg). They stick together after collision.

1
Calculate \( p_i \): \( p_i = m_1 v_{1i} + m_2 v_{2i} = (0.50)(2.0) + (0.50)(0) = \) 1.0 kg·m/s
2
Predict \( v_f \): \( (m_1 + m_2)v_f = p_i \), so \( v_f = \frac{1.0}{1.0} = \) 1.0 m/s
3
Lab result: Measured \( v_f = 0.96 \) m/s. Compute: \( p_f = (1.0)(0.96) = 0.96 \) kg·m/s
4
Percent difference: \( \frac{|1.0 - 0.96|}{1.0} \times 100\% = \) 4.0% (within acceptable error)

Example 2: Elastic Collision

Cart 1 (\( m_1 = 0.40 \) kg, \( v_{1i} = 3.0 \) m/s) hits Cart 2 (\( m_2 = 0.60 \) kg, at rest). Elastic bumpers are attached.

Example 3: Explosion (Separation)

Two carts start at rest and a spring-loaded plunger pushes them apart. \( m_1 = 0.30 \) kg, \( m_2 = 0.60 \) kg.

Before (at rest) \(m_1\) \(m_2\) \(p_i = 0\) → After (separated) \(m_1\) \(v_{1f}\) \(m_2\) \(v_{2f}\)
\(p_f = m_1 v_{1f} + m_2 v_{2f} = 0\)
Prediction: \(v_{1f}/v_{2f} = -m_2/m_1 = -2.0\)

Data Collection Best Practices

1
Multiple trials: Run at least 5 trials per condition to reduce random error and compute averages.
2
Record uncertainties: Note the precision of each instrument (mass balance: \( \pm 0.001 \) kg, photogate: \( \pm 0.001 \) s).
3
Organize data tables: Columns for trial number, \( m_1, m_2, v_{1i}, v_{2i}, v_{1f}, v_{2f}, p_i, p_f \), and percent difference.
4
Check for systematic error: If \( p_f \) is consistently lower than \( p_i \), friction may be the cause.

Handling Error Analysis

Random Error
Unpredictable fluctuations in measurement. Reduced by averaging multiple trials.
Systematic Error
Consistent bias (e.g., friction always removes momentum). Addressed by improving the setup.

Common Mistakes

✕ Ignoring Direction
Momentum is a vector. Forgetting to assign positive/negative signs to velocities in opposite directions makes calculations wrong.
✕ Unleveled Track
A tilted track means gravity adds an external force, violating the isolated system condition. Always level the track first.
✕ Single Trial Only
One trial can't distinguish a genuine result from random error. Always perform at least 5 trials and report the average.

Experiment Design Checklist

Step Action Purpose
1. Hypothesis \( \sum p_i = \sum p_f \) Testable prediction
2. Equipment Track, carts, sensors Measure \( m \) and \( v \)
3. Controls Level track, minimize friction Isolate the system
4. Data 5+ trials, record uncertainties Reduce random error
5. Analysis Compare \( p_i \) vs \( p_f \), \( \%\text{diff} \) Evaluate conservation
6. Conclusion Interpret results with error Support or refine hypothesis

Question 1 of 5

A student wants to test conservation of momentum using two carts on a track. Which variable must be controlled to ensure the system is isolated?

Question 2 of 5

In a perfectly inelastic collision, Cart 1 (\( m_1 = 1.0 \) kg, \( v_{1i} = 4.0 \) m/s) hits stationary Cart 2 (\( m_2 = 1.0 \) kg). What is the predicted final velocity?

Question 3 of 5

A student's 5 trials consistently show \( p_f \) about 8% lower than \( p_i \). What is the most likely explanation?

Question 4 of 5

In an explosion experiment, two objects initially at rest fly apart. Cart A (\( m = 0.20 \) kg) moves left at \( 3.0 \) m/s. Cart B has mass \( 0.30 \) kg. What is Cart B's velocity?

Question 5 of 5

Which improvement to an experiment would best reduce systematic error in a momentum conservation test?