Collision
Solve 1D two-body collisions: elastic final speeds, perfectly inelastic stick-together speed, or partial restitution with e. Momentum checks and kinetic-energy loss included. Free online calculator with formulas and step-by-step solutions.
Key formula
elastic v1,v2; inelastic v=(m1u1+m2u2)/(m1+m2); 0<=e<=1
How do you solve 1D collisions?
Conserve momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. For a perfectly inelastic collision the masses share one final speed v = (m₁u₁ + m₂u₂)/(m₁+m₂). For an elastic collision use the standard 1D formulas for v₁ and v₂ (equivalent to e = 1). For partial bounce, insert the restitution coefficient e into the general 1D formulas.
About this tool
A 1D collision calculator finds final speeds after two masses interact along a line. Elastic mode conserves kinetic energy; perfectly inelastic mode sticks the masses together; partial restitution uses a coefficient e between 0 and 1.
How to use this calculator
- Choose the variable you want to solve for and confirm which values are already known.
- Enter inputs in consistent units before calculating so the formula is applied correctly.
- Check whether the result is physically reasonable by comparing it with the size of the inputs.
- Use the step-by-step breakdown to see how the equation was rearranged and evaluated.
Worked examples
Example 1: equal masses, elastic
m₁ = m₂ = 2.0 kg, u₁ = 3.0 m/s, u₂ = 0. After an elastic collision, v₁ = 0 and v₂ = 3.0 m/s — the velocities swap. Momentum before and after is 6.0 kg·m/s.
Example 2: perfectly inelastic stick
m₁ = 3.0 kg at 4.0 m/s hits m₂ = 1.0 kg at rest and they stick. v = (3×4 + 1×0)/(3+1) = 3.0 m/s. KE falls from 24 J to 18 J.
Example 3: partial restitution e = 0.5
m₁ = 2.0 kg, m₂ = 2.0 kg, u₁ = 4.0 m/s, u₂ = 0, e = 0.5. Final speeds are between the elastic and sticky extremes; momentum still balances while KE is partly lost.
Frequently asked questions
What is conserved in every 1D collision?
Linear momentum is conserved when external forces are negligible during the short impact. Kinetic energy is conserved only for ideally elastic collisions (e = 1).
What is the formula for a perfectly inelastic collision?
The masses stick and move at one speed: v = (m₁u₁ + m₂u₂)/(m₁ + m₂). This case loses the maximum kinetic energy allowed by momentum conservation.
What are the elastic collision formulas for v₁ and v₂?
v₁ = (u₁(m₁ − m₂) + 2 m₂ u₂)/(m₁ + m₂) and v₂ = (u₂(m₂ − m₁) + 2 m₁ u₁)/(m₁ + m₂). For equal masses these swap the initial velocities.
What does the coefficient of restitution e mean?
e compares relative speed of separation to relative speed of approach. e = 1 is elastic, e = 0 is perfectly inelastic (stick), and 0 < e < 1 is a partial bounce.
How should I choose velocity signs?
Pick one direction as positive (usually right) and keep it for both masses before and after the collision. Opposing motion uses opposite signs.