Momentum Before and After Collision Calculator

Example Data Table

The following sample collisions illustrate different modes and typical scales.

Formula Used

Linear momentum p is defined as the product of mass and velocity: p = m × v. It is a vector quantity, sharing the velocity direction.

For a one-dimensional collision with two bodies, conservation of momentum states that the total momentum before collision equals the total momentum after collision:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

In a perfectly inelastic collision the bodies stick together after impact. Their common final velocity is:

v = (m₁u₁ + m₂u₂) / (m₁ + m₂)

In a perfectly elastic collision, both momentum and kinetic energy are conserved. The one-dimensional formulas for final velocities are:

v₁ = [(m₁ - m₂)u₁ + 2m₂u₂] / (m₁ + m₂)

v₂ = [(m₂ - m₁)u₂ + 2m₁u₁] / (m₁ + m₂)

For a general one-dimensional collision with coefficient of restitution e, the relative speed relation is v₂ - v₁ = -e(u₂ - u₁). Solving jointly with momentum conservation yields the custom restitution formulas implemented above.

The calculator also evaluates center of mass velocities and impulses on each body to give a deeper picture of collision dynamics.

How to Use This Calculator

1. Choose the collision mode. Use Check conservation when you already know all initial and final velocities and simply want to verify momentum.
2. Select Perfectly elastic or Perfectly inelastic to generate final velocities that satisfy standard textbook models for collisions.
3. Select Custom restitution when you have a measured or assumed coefficient of restitution e, then enter the value to compute partially elastic outcomes.
4. Enter masses for both bodies and select appropriate units. Units are internally converted to kilograms for consistent calculations.
5. Enter the initial velocities. You can work in metres per second, kilometres per hour, or feet per second depending on your experiment or example.
6. For the check mode, also enter final velocities. For other modes, the calculator will determine final velocities automatically.
7. Press Calculate to compute momentum, center of mass velocities, impulses, and kinetic energies, then export everything to CSV or PDF for documentation.

Understanding Momentum Before and After Collision

Overview of Collision Momentum

Momentum before and after a collision describes how motion is redistributed between interacting bodies. This calculator focuses on straight-line impacts, so the momentum values are signed along one axis only. That convention lets you treat opposite directions as negative and positive values instead of separate dimensions. It also aligns naturally with typical track and rail experiments.

Role of Mass and Velocity

Because momentum equals mass times velocity, heavier objects contribute strongly to total momentum. A light cart moving quickly can still match the momentum of a heavier cart moving slowly in the opposite direction. The input fields for masses and speeds let you explore those trade-offs very systematically.

Different Collision Types

Elastic collisions conserve both momentum and kinetic energy, making them useful for idealized problems and demonstrations. Perfectly inelastic collisions conserve momentum but lose kinetic energy as heat, sound, or deformation. Real-world collisions usually sit between those extremes, which you can mimic with the custom restitution mode.

Using Coefficient of Restitution

The coefficient of restitution e measures how “bouncy” a collision is. When e equals one, the collision is perfectly elastic. When e is zero, the collision is perfectly inelastic, with objects moving together afterward. Intermediate values approximate rubber balls, bumpers, or other partially elastic materials.

Center of Mass Perspective

Viewing the collision from the center of mass frame simplifies interpretation. In that frame, total momentum is always zero. Before and after collision, velocities simply reverse direction when the interaction is perfectly elastic. The calculator reports center of mass speed to highlight that physically important quantity.

Impulses During Impact

The change in momentum of each body during the collision is called impulse. It equals the average contact force multiplied by the interaction time. Equal and opposite impulses reflect Newton’s third law between the two bodies. The impulse values calculated here connect abstract momentum changes to measurable forces. This helps bridge theoretical lessons with real sensor readings in labs.

Practical Applications

You can apply this calculator to carts on tracks, colliding pucks, or simplified vehicle impacts. It is equally useful for homework, laboratory experiments, and quick design estimates. By comparing percentage momentum change and kinetic energy change, you quickly see how realistic your measurements or textbook parameters are. That insight supports better experimental planning and clearer classroom discussions.

Frequently Asked Questions

What does a positive or negative velocity mean here?

Positive and negative velocities simply represent opposite directions along one line. You can choose which direction is positive. The important part is staying consistent when entering all speeds before and after the collision.

Can this calculator handle more than two bodies colliding?

The calculator is designed specifically for two-body, one-dimensional collisions. Multi-body events can often be treated as a sequence of pairwise collisions, but you would need to analyze each stage separately using appropriate intermediate velocities.

Why is kinetic energy not always conserved in collisions?

In many real impacts, some kinetic energy transforms into heat, sound, vibration, or permanent deformation. Momentum still remains conserved for the isolated system, but kinetic energy decreases, especially when materials are soft, crumple, or stick together during the collision.

What range of coefficient of restitution should I use?

Typical values lie between zero and one, where zero means perfectly inelastic and one means perfectly elastic. Real materials usually fall in between. Measured values slightly above one sometimes appear from experimental noise and should be interpreted cautiously.

How accurate are results when I mix different unit choices?

Internally, the calculator converts all masses to kilograms and velocities to metres per second. As long as you choose the correct units from the menus, the final momentum and energy values are consistent and directly comparable between scenarios.

Can I use this tool for safety or legal crash analysis?

This calculator is intended for education and preliminary insight, not certified engineering or forensic work. Professional crash reconstruction requires detailed models, high-quality measurements, and regulatory standards that go beyond simple one-dimensional momentum calculations.