Hong Life CalculatorsUnderstanding the collision calculator — concept and principle
When two objects collide head-on along a line, their velocities afterwards are fixed by two facts: total momentum is conserved, and the coefficient of restitution e describes how "bouncy" the impact is. This calculator gives both final velocities and the kinetic energy lost, for anything from a perfectly elastic collision (e = 1) to a perfectly inelastic one where the bodies stick together (e = 0). It is used for physics problems, billiards and Newton's cradle intuition, and simple crash analysis.
How it is calculated
Masses m₁, m₂ in kg; initial velocities u₁, u₂ in m/s (signs give direction, so −1 means moving left):
- v₁ = (m₁u₁ + m₂u₂ − m₂ e (u₁ − u₂)) ÷ (m₁ + m₂)
- v₂ = (m₁u₁ + m₂u₂ + m₁ e (u₁ − u₂)) ÷ (m₁ + m₂)
- KE lost = ½ · m₁m₂ ÷ (m₁ + m₂) · (1 − e²) · (u₁ − u₂)², in joules.
The principle
With no external force along the line, momentum before equals momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. That one equation has two unknowns, so Newton's experimental law of restitution supplies the second: the separation speed equals e times the approach speed, v₂ − v₁ = e (u₁ − u₂). Solving the pair gives the formulas above. The energy loss depends only on the reduced mass m₁m₂ ÷ (m₁ + m₂) and the relative speed; e = 1 conserves kinetic energy, and e = 0 loses the most energy that momentum conservation allows.
Worked example
- Inputs: m₁ = 2 kg, u₁ = 3 m/s, m₂ = 1 kg, u₂ = −1 m/s, e = 1.
- Total momentum: 2 × 3 + 1 × (−1) = 5 kg·m/s; approach speed u₁ − u₂ = 4 m/s.
- v₁ = (5 − 1 × 1 × 4) ÷ 3 ≈ 0.3333 m/s.
- v₂ = (5 + 2 × 1 × 4) ÷ 3 ≈ 4.3333 m/s.
- KE lost = ½ × (2/3) × (1 − 1) × 16 = 0 J. Check: KE is 9.5 J before and after.
| m₁ (kg) | v₁ after (m/s) |
|---|---|
| 0.5 | -2.333333333 |
| 1 | -1 |
| 1.5 | -0.2 |
| 2 | 0.3333333333 |
| 2.5 | 0.7142857143 |
| 3 | 1 |
| 4 | 1.4 |
How to read the result
Positive and negative signs show direction after the impact. Here the heavier ball keeps moving slowly forward while the light one rebounds at 4.33 m/s. Typical e values are close to 1 for hardened steel or billiard balls, a mid-range value for a well-inflated sports ball, and near 0 for clay or vehicles in a crash. With e = 0 and the same inputs, both move at 5/3 ≈ 1.667 m/s together and about 5.33 J becomes heat, sound and deformation.
Common mistakes
- Entering speeds without signs for objects moving towards each other.
- Expecting kinetic energy to be conserved in every collision; only momentum always is.
- Using e above 1, which would require extra energy, as in an explosion.
Frequently asked questions
What happens when equal masses collide elastically?
They exchange velocities. A moving ball hitting an identical resting ball stops, and the other moves off at the original speed.
How do I measure the coefficient of restitution?
Drop a ball on a hard floor: e = √(rebound height ÷ drop height).
Where does the lost kinetic energy go?
Into permanent deformation, heat, sound and vibration.
Sources: Isaac Newton, Philosophiæ Naturalis Principia Mathematica, 1687 (Laws of Motion, Corollaries and Scholium) · Halliday, Resnick and Walker, Fundamentals of Physics, Wiley