Calculate the missing mass or initial/final velocity in a one-dimensional elastic collision using two objects’ masses and velocities.

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Perfectly elastic, one-dimensional Newtonian model. Use one positive direction and signed velocities; speeds must be well below light speed.

Elastic Collision Formula

For a one-dimensional perfectly elastic collision, both momentum and kinetic energy are conserved. The calculator uses the following equations, with all values converted to base units before solving.

v₁ = (m₁ - m₂) / (m₁ + m₂)u₁ + (2m₂) / (m₁ + m₂)u₂
v₂ = (2m₁) / (m₁ + m₂)u₁ + (m₂ - m₁) / (m₁ + m₂)u₂
u₁ = (m₁ - m₂) / (m₁ + m₂)v₁ + (2m₂) / (m₁ + m₂)v₂
u₂ = (2m₁) / (m₁ + m₂)v₁ + (m₂ - m₁) / (m₁ + m₂)v₂
m₁ = (m₂(v₂ - u₂)) / (u₁ - v₁)
m₂ = (m₁(u₁ - v₁)) / (v₂ - u₂)
  • m1 = mass of object 1
  • m2 = mass of object 2
  • u1 = initial velocity of object 1
  • u2 = initial velocity of object 2
  • v1 = final velocity of object 1
  • v2 = final velocity of object 2

Choose the value to solve. The both-velocity modes need two masses and two velocities. Single-value modes preserve the five-known-value solves and check the other supplied velocity for consistency. Mass solves rearrange momentum and then verify the elastic velocity equations; inconsistent or nonunique inputs are rejected.

Common Elastic Collision Patterns

Situation Typical result What it means
Equal masses, object 2 starts at rest Object 1 stops, object 2 takes object 1’s speed The moving object transfers its velocity to the other object.
Object 1 is much lighter than object 2 Object 1 rebounds strongly A light object bouncing off a heavy object reverses direction.
Object 1 is much heavier than object 2 Object 1 changes little, object 2 moves away quickly The heavier object keeps most of its motion.
Objects move toward each other One velocity should be entered as negative Velocity signs show direction in a one-dimensional setup.

Supported Unit Conversions

Quantity Input units Base unit used for calculation
Mass kg, lb kg
Velocity m/s, ft/s m/s

Elastic Collision Examples

Example 1: Find both final velocities for equal masses

Suppose object 1 has a mass of 2 kg and starts at 6 m/s. Object 2 has a mass of 2 kg and starts at 0 m/s.

  • m1 = 2 kg
  • u1 = 6 m/s
  • m2 = 2 kg
  • u2 = 0 m/s

For equal masses where the second object starts at rest, the velocities exchange. The final velocity of object 1 is 0 m/s, and the final velocity of object 2 is 6 m/s.

Example 2: Find final velocities for unequal masses

Suppose object 1 has a mass of 3 kg and starts at 4 m/s. Object 2 has a mass of 1 kg and starts at 0 m/s.

v₁ = (3 - 1) / (3 + 1)(4) + (2(1)) / (3 + 1)(0) = 2 m / s
v₂ = (2(3)) / (3 + 1)(4) + (1 - 3) / (3 + 1)(0) = 6 m / s

The final velocity of object 1 is 2 m/s, and the final velocity of object 2 is 6 m/s.

Elastic Collision FAQ

Can velocity be negative in an elastic collision?

Yes. Velocity is directional. In a one-dimensional collision, choose one direction as positive and enter motion in the opposite direction as negative. For example, if object 1 moves right at 5 m/s and object 2 moves left at 3 m/s, you can enter 5 m/s and -3 m/s.

What makes a collision elastic?

A collision is elastic when total momentum and total kinetic energy are conserved. In real life, many collisions lose some kinetic energy to heat, sound, deformation, or rotation. The calculator assumes a perfectly elastic, one-dimensional collision.

How many known values does each solve need?

Choose both final or both initial velocities for a four-value solve. A single missing mass or velocity uses five known values, with a consistency check. Only fields relevant to the selected task are shown.