Elastic Collision Calculator
Calculate final velocities after a perfectly elastic 1D collision and verify conservation of momentum and kinetic energy.
Calculator is for informational purposes only. Terms and Conditions
Elastic collision equations
For a perfectly elastic 1D collision, both momentum and kinetic energy are conserved.
Choose what to solve for
Pick the collision setup, solve mode, and unit preferences. The visible inputs update automatically.
Enter the known values
Enter velocity magnitudes as positive numbers, then choose the direction. The calculator handles signs internally.
Before and after collision visual
Ball size represents relative mass. Arrow length and direction represent velocity.
Solution
Final velocities are calculated using conservation of momentum and kinetic energy.
Quick checks
- Object 1 final velocity—
- Object 2 final velocity—
- Momentum difference—
- Kinetic energy difference—
- Relative speed check—
- Center of mass velocity—
Source, standards, and assumptions
This calculator uses the standard educational mechanics model for a perfectly elastic one-dimensional collision.
Calculation basis: Conservation of linear momentum and conservation of kinetic energy for a perfectly elastic 1D collision.
- No single governing design code standard is required for this simplified educational physics calculation.
- Coefficient of restitution is assumed to be e = 1.0.
- Objects are treated as point masses moving along one straight collision line.
- External forces during the short collision interval are ignored.
- Rotational energy, deformation losses, friction losses, and sound/heat losses are ignored.
- Standard unit conversion constants are used for mass, velocity, energy, and momentum.
Solution steps
Review the substitution, conservation checks, and interpretation.
Show calculation steps
- Enter values to see the full calculation steps and conservation checks.
Warnings
How to Calculate an Elastic Collision Correctly
Use this Elastic Collision Calculator to solve final velocities after a perfectly elastic one-dimensional collision. The calculator is built for the values users usually need most: object masses, initial velocities, velocity direction, final velocities, momentum checks, and kinetic energy checks.
The article below explains how to use the calculator, what each variable means, how the formulas work, how to handle negative velocity, and how to check whether the result makes physical sense. It is designed for students, engineering learners, physics homework problems, and anyone trying to understand a two-object elastic collision.
What Is an Elastic Collision?
An elastic collision is a collision where total momentum and total kinetic energy are conserved. In a perfectly elastic collision, the objects may bounce apart, exchange velocities, or continue moving in the same direction depending on their masses and incoming velocities.
In real life, most collisions are not perfectly elastic because some energy is usually converted into sound, heat, permanent deformation, vibration, or rotation. This calculator uses the ideal physics model where the coefficient of restitution is effectively e = 1.
Direct answer
In a perfectly elastic collision, momentum is conserved and kinetic energy is conserved. That is why this calculator reports both final velocities and conservation checks.
How to Use the Elastic Collision Calculator
Most users searching for an elastic collision calculator want to know what happens after two objects collide. The calculator above is designed to answer that quickly while still showing enough physics detail to verify the result.
Choose the collision setup
Select whether both objects may be moving or whether object 2 starts at rest. The stationary-target option is useful for common cart, ball, puck, and billiard-style problems.
Enter each mass
Enter the mass of object 1 and object 2. The calculator supports common units such as kilograms, grams, pounds mass, and slugs, then converts internally for the calculation.
Enter the initial velocities and directions
Use the velocity magnitude field for speed and the direction selector for sign. A velocity to the right is usually positive, while a velocity to the left is usually negative.
Read the final velocities
The result gives the final velocity of object 1 and object 2. A negative final velocity means the object moves in the negative direction after the collision.
Check momentum and kinetic energy
For a perfectly elastic collision, the before-and-after momentum and kinetic energy values should match apart from rounding.
Important calculator assumption
This calculator assumes a one-dimensional collision along a straight line. For a two-dimensional collision, you need velocity components and the collision normal direction.
Elastic Collision Formula
For a one-dimensional perfectly elastic collision between two objects, the final velocities can be calculated directly from the two masses and two initial velocities.
Final velocity of object 1
Use this equation to find the final velocity of object 1 after the elastic collision.
Final velocity of object 2
Use this equation to find the final velocity of object 2 after the elastic collision.
These equations come from applying conservation of momentum and conservation of kinetic energy to a two-object collision. They are most appropriate when the objects move along the same line before and after impact.
What the Elastic Collision Variables Mean
Before using the formulas, make sure each variable is understood correctly. The most common mistakes come from mixing up initial and final velocity or entering the wrong sign for direction.
| Symbol | Meaning | What to Enter or Read |
|---|---|---|
| m1 | Mass of object 1 | The mass of the first object, such as the moving cart, ball, puck, or particle |
| m2 | Mass of object 2 | The mass of the second object involved in the collision |
| v1i | Initial velocity of object 1 | The velocity of object 1 before collision, including direction |
| v2i | Initial velocity of object 2 | The velocity of object 2 before collision, including direction |
| v1f | Final velocity of object 1 | The calculated velocity of object 1 after collision |
| v2f | Final velocity of object 2 | The calculated velocity of object 2 after collision |
Velocity is a signed value. The magnitude tells you how fast the object moves, while the sign tells you the direction along the chosen collision axis.
How to Choose Positive and Negative Velocity
Direction is one of the most important parts of an elastic collision problem. A calculator cannot know which way is positive unless you define it. A simple approach is to call motion to the right positive and motion to the left negative.
| Motion | Typical Sign | How to Interpret It |
|---|---|---|
| Object moves right | Positive | The object is moving in the chosen positive direction |
| Object moves left | Negative | The object is moving opposite the chosen positive direction |
| Object stops | Zero | The object has no velocity after the collision |
| Object rebounds | Often changes sign | A positive incoming velocity may become a negative final velocity |
Negative velocity is not an error
A negative final velocity simply means the object is moving in the negative direction after collision. It does not mean the speed is physically impossible.
Step-by-Step Elastic Collision Example
A worked example is the easiest way to see what the calculator is doing. In this example, object 1 moves toward object 2, which is initially at rest.
Calculate object 1 final velocity
Calculate object 2 final velocity
Result
Object 1 rebounds at 1 m/s in the negative direction. Object 2 moves forward at 4 m/s.
Check the answer
Initial momentum is \(2(5)+3(0)=10\text{ kg·m/s}\). Final momentum is \(2(-1)+3(4)=10\text{ kg·m/s}\). Initial kinetic energy is \(25\text{ J}\). Final kinetic energy is \(1+24=25\text{ J}\). Both momentum and kinetic energy are conserved, so the result is consistent with a perfectly elastic collision.
Special Case: One Object Starts at Rest
Many elastic collision problems involve a moving object striking a stationary object. In that case, \(v_{2i}=0\), so the equations simplify.
Object 1 final velocity when object 2 starts at rest
Object 2 final velocity when object 2 starts at rest
| Mass Relationship | Typical Result | Physical Interpretation |
|---|---|---|
| m₁ = m₂ | Object 1 stops and object 2 takes object 1’s velocity | Equal masses exchange velocities |
| m₁ > m₂ | Object 1 continues forward and object 2 moves faster | The heavier object keeps moving after impact |
| m₁ < m₂ | Object 1 rebounds and object 2 moves forward | The lighter object bounces back from the heavier object |
| m₂ is much larger than m₁ | Object 1 nearly reverses direction | The collision behaves like a bounce from a massive wall |
Special Case: Equal Masses
One of the most useful shortcuts in elastic collision problems occurs when the two masses are equal. In a one-dimensional perfectly elastic collision, equal masses exchange velocities.
Equal-mass shortcut
If \(m_1=m_2\), then object 1 leaves with object 2’s original velocity and object 2 leaves with object 1’s original velocity.
Equal-mass result
This is why a row of nearly identical balls or carts can appear to transfer motion from one object to the next. The ideal formula assumes no energy loss, no rotation, and a straight-line collision.
Why the Calculator Checks Momentum and Kinetic Energy
The final velocities are not the only important results. A good elastic collision calculator should also verify whether the output satisfies the two conservation rules that define the ideal model.
Momentum
Total momentum before and after the collision should match in an isolated system.
Kinetic energy
Total kinetic energy before and after the collision should match for a perfectly elastic collision.
Momentum check
Confirms the collision result is consistent with conservation of linear momentum.
Energy check
Confirms the result is elastic instead of inelastic.
Relative speed check
For a perfectly elastic 1D collision, separation speed equals approach speed.
Elastic vs. Inelastic Collision
Elastic and inelastic collisions are often confused. Momentum can be conserved in both types if the system is isolated, but kinetic energy is only conserved in a perfectly elastic collision.
| Collision Type | Momentum Conserved? | Kinetic Energy Conserved? | Typical Behavior |
|---|---|---|---|
| Perfectly elastic | Yes, in an isolated system | Yes | Objects bounce without net kinetic energy loss |
| Inelastic | Yes, in an isolated system | No | Some kinetic energy becomes heat, sound, deformation, or rotation |
| Perfectly inelastic | Yes, in an isolated system | No | Objects stick together after collision |
This calculator is for the first case: a perfectly elastic collision. If objects stick together, deform heavily, or lose significant energy, use an inelastic collision method instead.
Limitations of the Elastic Collision Model
A calculator can return a correct ideal-physics answer that does not perfectly describe a real impact. Use the result as an educational mechanics calculation, not as a complete crash, impact, or structural safety model.
One-dimensional motion
The calculator assumes both objects move along the same straight line before and after impact.
No energy loss
The model assumes no kinetic energy is lost to sound, heat, deformation, vibration, or rotation.
No external impulse
External forces during the short impact interval are ignored.
No shape or contact details
The model does not calculate stress, deformation, contact time, or collision force.
When this calculator is not enough
Do not use this ideal elastic collision result by itself for vehicle crashes, safety-critical impact design, sports injury analysis, structural impact loading, or any situation where deformation and force history matter.
Common Elastic Collision Mistakes That Cause Wrong Answers
Most wrong elastic collision answers come from sign convention, unit mistakes, or applying an elastic model to a collision that is not actually elastic.
Common Don’ts
- Enter negative velocity without first defining which direction is positive
- Treat speed and velocity as the same thing
- Forget that kinetic energy uses velocity squared
- Use the elastic formula for objects that stick together
- Ignore the meaning of a negative final velocity
- Mix mass or velocity units without converting them
Better Checks
- Pick a positive direction before entering velocities
- Use signed velocity for direction and speed magnitude for size
- Check momentum before and after the collision
- Check kinetic energy before and after the collision
- Use the stationary-target shortcut when \(v_{2i}=0\)
- Interpret negative results as direction, not failure
Frequently Asked Questions
What is conserved in an elastic collision?
In a perfectly elastic collision, total momentum and total kinetic energy are conserved. This is why the calculator reports both momentum and kinetic energy checks.
What is the elastic collision formula?
For a 1D perfectly elastic collision, \(v_{1f}=\frac{m_1-m_2}{m_1+m_2}v_{1i}+\frac{2m_2}{m_1+m_2}v_{2i}\) and \(v_{2f}=\frac{2m_1}{m_1+m_2}v_{1i}+\frac{m_2-m_1}{m_1+m_2}v_{2i}\).
What happens when two equal masses collide elastically?
In a one-dimensional perfectly elastic collision between equal masses, the objects exchange velocities. If object 2 is initially at rest, object 1 stops and object 2 moves away with object 1’s original velocity.
What if one object starts at rest?
Set the initial velocity of that object to zero. If object 2 starts at rest, the simplified formulas are \(v_{1f}=\frac{m_1-m_2}{m_1+m_2}v_{1i}\) and \(v_{2f}=\frac{2m_1}{m_1+m_2}v_{1i}\).
Why did I get a negative final velocity?
A negative final velocity means the object moves in the negative direction after the collision. It does not mean the answer is wrong or physically impossible.
Is kinetic energy always conserved in collisions?
No. Kinetic energy is conserved in a perfectly elastic collision, but it is not conserved in inelastic or perfectly inelastic collisions.
Is this calculator for 2D collisions?
This calculator is for one-dimensional elastic collisions. A two-dimensional elastic collision requires velocity components and the collision normal direction.
Can I use this for real-world crashes?
Not by itself. Real crashes often involve deformation, heat, sound, friction, rotation, and other losses. This calculator is best for idealized physics and educational collision problems.