Elastic Collision Calculator
Calculate both final velocities after a one-dimensional perfectly elastic collision and verify momentum and kinetic-energy conservation.
Calculator is for informational purposes only. Terms and Conditions
The model assumes an isolated two-object collision along one line with no rotational, deformation, or external-force effects during impact; positive velocity is to the right and negative velocity is to the left.
Choose display settings
Changing units converts the existing physical quantities instead of reinterpreting the numbers.
Enter masses and initial velocities
Use positive velocity for motion to the right and negative velocity for motion to the left.
All four fields are required. Example values are loaded initially; edit any field to use your own collision.
Final velocities
Both answers are shown first, followed by conservation checks and physical interpretation.
Conservation and collision checks
- Momentum before / after—
- Kinetic energy before / after—
Show calculation steps Review inputs, equations, substitutions, conservation checks, impulse, and center-of-mass behavior
- Enter valid values to see the complete calculation.
Collision and kinetic-energy view
The diagram shows direction before and after impact; object size is normalized from mass and is not a physical scale drawing.
- Enter valid values to populate the kinetic-energy comparison.
Method, Sources, and Assumptions
Calculation basis, authoritative reference, sign convention, and model limitations.
Final velocities are calculated from simultaneous conservation of linear momentum and kinetic energy for a one-dimensional perfectly elastic two-body collision.
- Positive velocity is defined as rightward; negative velocity is leftward.
- Momentum and total kinetic energy are conserved through the modeled collision.
- The model excludes deformation energy, rotation, frictional impulse, contact duration, stress, and collision force.
- The collision diagram is normalized for readability and is not drawn to physical mass, size, distance, or time scale.
Calculator guide
How to Read Your Elastic Collision Result
The calculator above finds the final velocities of two objects after a one-dimensional perfectly elastic collision from four inputs: Object 1 mass, Object 1 initial velocity, Object 2 mass, and Object 2 initial velocity. It also checks whether total momentum and total kinetic energy are conserved, then reports direction, center-of-mass velocity, impulse, and the kinetic-energy distribution.
Use a consistent sign convention: positive velocity means motion to the right and negative velocity means motion to the left. A negative final velocity is not a negative speed; it means that object is moving in the negative direction after the collision. The model applies to two objects moving along one line and assumes the collision is perfectly elastic.
- Inputs
- Two positive masses and two signed initial velocities
- Main output
- Final velocity and direction of each object
- Core check
- Momentum and kinetic energy must both match before and after
Elastic Collision Equations and Variables
A perfectly elastic collision conserves both linear momentum and kinetic energy. Combining those two conservation laws gives a closed-form equation for each final velocity, so the calculator can solve both outputs directly from the two masses and two initial velocities.
Final velocity of Object 1
Object 1’s final velocity depends on both initial velocities and on how its mass compares with Object 2’s mass.
Final velocity of Object 2
Object 2’s final velocity is found from the same masses and initial velocities using the companion elastic-collision equation.
Conservation checks
In plain language, the system must have the same total momentum and the same total kinetic energy immediately before and immediately after the idealized collision.
- \(m_1\)
- Mass of Object 1. It must be positive.
- \(m_2\)
- Mass of Object 2. It must be positive.
- \(u_1\)
- Initial velocity of Object 1, including its direction by sign.
- \(u_2\)
- Initial velocity of Object 2, including its direction by sign.
- \(v_{1f}\)
- Final velocity of Object 1 after the collision.
- \(v_{2f}\)
- Final velocity of Object 2 after the collision.
Worked Elastic Collision Example
Use the calculator’s default example: a \(2\text{ kg}\) Object 1 moving right at \(5\text{ m/s}\) strikes a stationary \(3\text{ kg}\) Object 2. This case is useful because the lighter incoming object rebounds, so it also demonstrates what a negative final velocity means.
Substitute for Object 1
Substitute for Object 2
Result
Object 1: \(-1\text{ m/s}\) | Object 2: \(4\text{ m/s}\)
Object 1 reverses direction and moves left at \(1\text{ m/s}\). Object 2 moves right at \(4\text{ m/s}\).
How Mass and Direction Change the Outcome
The final velocities are controlled by the mass ratio and the signed initial velocities. For the common stationary-target case \(u_2=0\), the sign of \(m_1-m_2\) immediately tells you whether Object 1 continues forward, stops, or rebounds.
| Mass relationship | Object 1 after collision | Object 2 after collision |
|---|---|---|
| \(m_1<m_2\) | Rebounds; \(v_{1f}<0\) | Moves forward |
| \(m_1=m_2\) | Stops; \(v_{1f}=0\) | Takes Object 1’s initial velocity |
| \(m_1>m_2\) | Continues forward | Moves forward and can leave faster than Object 1’s incoming speed |
Equal masses exchange velocities
When \(m_1=m_2\), the equations reduce to \(v_{1f}=u_2\) and \(v_{2f}=u_1\). If Object 2 starts at rest, Object 1 stops and Object 2 leaves with Object 1’s original velocity.
Energy can move between objects
Total kinetic energy stays constant, but each object’s share can change dramatically. With a stationary target, complete transfer of the incoming object’s kinetic energy occurs in the ideal equal-mass case.
Center-of-mass velocity stays fixed
For the isolated two-object system, the center-of-mass velocity \(V_{CM}=(m_1u_1+m_2u_2)/(m_1+m_2)\) does not change during the internal collision. The calculator reports this as an additional consistency check.
Common Elastic Collision Mistakes
Most wrong answers come from sign conventions, unit mismatches, or using the elastic equations for a collision that does not meet the model assumptions.
Dropping the negative sign
A negative final velocity usually means the object reversed direction. Converting \(-1\text{ m/s}\) to \(+1\text{ m/s}\) changes the momentum balance and produces the wrong physical interpretation.
Mixing speed and velocity
Speed is a nonnegative magnitude; velocity includes direction. Use signed velocities in the collision equations and interpret the absolute value only when you want the speed.
Using the wrong unit label
The calculator converts supported mass and velocity units internally. When checking the work by hand, convert quantities consistently before substitution rather than multiplying, for example, pounds directly by meters per second and calling the result SI momentum.
Assuming every real impact is perfectly elastic
Real macroscopic impacts usually convert some kinetic energy into deformation, sound, heat, rotation, or other forms. If total kinetic energy is not conserved, a perfectly elastic model is not the right description.
Assumptions and Limits of the Model
This is an exact classical-mechanics solution for its stated idealized model, not a general impact-analysis tool. The result is reliable only when the physical problem can reasonably be represented as a two-body, one-dimensional, perfectly elastic collision.
One-dimensional motion
All motion is treated along one signed axis. A general two-dimensional collision requires velocity components and additional directional or contact information.
Perfectly elastic interaction
Total kinetic energy is assumed to be conserved. In coefficient-of-restitution language, the ideal perfectly elastic case corresponds to \(e=1\).
Isolated two-object system
The collision calculation assumes external impulse is negligible over the impact interval, so the system’s total linear momentum is conserved.
No contact mechanics
The calculator does not determine deformation, stress, contact time, peak force, frictional impulse, rotational effects, or material damage.
Elastic Collision FAQ
These questions address the most common interpretation and boundary cases that are not obvious from the final velocities alone.
What is conserved in a perfectly elastic collision?
For the isolated two-object model used here, total linear momentum and total kinetic energy are both conserved. Individual objects can gain or lose momentum and kinetic energy while the system totals remain unchanged.
What happens when two equal masses collide elastically?
In one dimension, equal masses exchange velocities: \(v_{1f}=u_2\) and \(v_{2f}=u_1\). If Object 2 is initially stationary, Object 1 stops and Object 2 leaves with Object 1’s original velocity.
Why is a final velocity negative?
A negative velocity indicates motion in the negative direction selected by the sign convention. If right is positive, a negative result means the object moves left after the collision; its speed is the absolute value of that velocity.
Can both objects move in the same direction after an elastic collision?
Yes. For example, when a heavier moving object strikes a lighter stationary object in one dimension, both can move forward after the collision while still conserving total momentum and kinetic energy.
Is kinetic energy conserved in every collision?
No. Total kinetic energy is conserved in an elastic collision. In an inelastic collision, momentum can still be conserved for an isolated system while some kinetic energy is transformed into other forms.
Can this calculator find collision force?
No. The change in momentum gives impulse, but force also depends on how that impulse is distributed over time. Contact duration, deformation, stiffness, material behavior, and other impact details are outside this calculator’s model.
Can this calculator solve a two-dimensional collision?
No. This calculator solves one-dimensional motion along a single signed axis. A general two-dimensional collision needs momentum components and additional information about directions, geometry, or the line of impact.