Elastic Collision (1D) Calculator

How to Use the Calculator

1. Enter Mass 1 (m₁) and select its unit (kg or g).
2. Enter Initial Velocity 1 (u₁) and choose m/s or km/h.
3. Enter Mass 2 (m₂) with its unit.
4. Enter Initial Velocity 2 (u₂) with its unit. Use a negative value if the second object moves in the opposite direction.
5. Read the results: the Final Velocity of Object 1 (v₁) and the Final Velocity of Object 2 (v₂), both in m/s.

All inputs are converted internally to kilograms and meters per second, so you can mix units freely — for example, one mass in grams and one in kilograms — and the results will still be correct.

What Is an Elastic Collision?

A collision is called elastic when both momentum and kinetic energy are conserved. In other words, no kinetic energy is lost to heat, sound, deformation, or vibration — the total energy of motion before the impact equals the total energy of motion after it. This is an idealization: real collisions always lose some energy, but hard, springy objects such as steel balls or billiard balls come very close to the ideal.

The calculator handles one-dimensional (1D) collisions, meaning both objects move along the same straight line, either in the same direction or toward each other. Because the motion is confined to one axis, the outcome is fully described by two numbers: the final velocity of each object. A negative result simply means the object moves in the negative direction after the collision.

Some special cases are worth recognizing. If two objects of equal mass collide elastically, they simply exchange velocities. If a moving object hits a much heavier object at rest, it bounces back with nearly the same speed. If it hits a much lighter object at rest, it barely slows down while the light object shoots off at up to twice the incoming speed — which is exactly what happens when a golf club strikes a ball.

Interpreting the Results

The sign of each final velocity tells you the direction of motion. In the default example above, both values are positive, so both objects move in the original direction of object 1. If a result is negative, that object rebounds and travels the opposite way. You can verify that momentum is conserved by checking that \(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\) — a quick way to confirm your inputs were entered correctly.

Assumptions and Limitations

• Perfectly elastic: no energy is lost to heat, sound, or deformation. Real collisions are at best close to elastic; for problems involving energy loss, a coefficient of restitution is needed instead.
• One dimension only: all motion is along a single line. Two-dimensional collisions (such as glancing billiard-ball impacts) require vector components and are outside the scope of this tool.
• No external forces: friction, gravity along the motion path, and other impulses during the brief collision are neglected — a good approximation because collisions last only milliseconds.
• Point masses or non-rotating bodies: rotational motion is ignored. For rotating systems, see the Angular Momentum Calculator.
• Direction convention: velocities must be signed. If both objects approach each other, one of the initial velocities must be entered as negative.

Practical Tips

Elastic collision problems appear throughout introductory mechanics — Newton's cradle, air-track experiments, gas molecule collisions, and particle scattering all rely on the same equations. When solving problems by hand, always set up a clear sign convention first (usually right = positive) and stick to it for every velocity. After the collision, the average speed of each object over a known distance, or its subsequent free fall if it leaves a table or track, can be found with our other physics calculators, including the Projectile Motion Calculator.