Elastic Collision Calculator
Calculate post-collision velocities in a 1D elastic collision.
Enter both masses and initial velocities to find final speeds and verify energy conservation.
A perfectly elastic collision conserves both momentum and kinetic energy. No energy is lost to heat, deformation, or sound. Real-world collisions between billiard balls, gas molecules, and subatomic particles come close to this ideal.
Two conservation laws constrain the outcome simultaneously:
Conservation of momentum: m1v1 + m2v2 = m1v1’ + m2v2' Conservation of kinetic energy: (1/2)m1v1² + (1/2)m2v2² = (1/2)m1v1’² + (1/2)m2v2’²
Solving these two equations simultaneously for the final velocities:
v1’ = ((m1 - m2) x v1 + 2 x m2 x v2) / (m1 + m2) v2’ = ((m2 - m1) x v2 + 2 x m1 x v1) / (m1 + m2)
Special cases worth knowing:
- Equal masses (m1 = m2): the objects exchange velocities completely. Object 1 stops; object 2 moves at object 1’s original speed. This is why Newton’s cradle works.
- Very heavy hitting very light (m1 » m2): the heavy object barely slows down; the light object flies off at nearly twice the heavy object’s speed.
- Very light hitting very heavy (m1 « m2): the light object bounces back at nearly the same speed; the heavy object barely moves.
A worked example. A 2 kg cart moving right at 5 m/s meets a 3 kg cart moving left at 2 m/s.
v1’ = ((2 − 3) × 5 + 2 × 3 × (−2)) / 5 = (−5 − 12) / 5 = −3.4 m/s v2’ = ((3 − 2) × (−2) + 2 × 2 × 5) / 5 = (−2 + 20) / 5 = +3.6 m/s
Both carts reverse direction. Check the books: momentum before is 2(5) + 3(−2) = 4 kg·m/s, and after it is 2(−3.4) + 3(3.6) = −6.8 + 10.8 = 4. Kinetic energy before is 25 + 6 = 31 J, and after it is 11.56 + 19.44 = 31 J. Both hold.
There is a shortcut for checking any elastic result. The relative speed of approach must equal the relative speed of separation: here 5 − (−2) = 7 m/s closing, and 3.6 − (−3.4) = 7 m/s separating. If those two do not match, the answer is wrong, and you did not need the energy equation to find out.
Enter positive velocity for rightward motion and negative for leftward. The calculator prints the momentum and energy residuals so you can see the conservation laws holding rather than take them on trust.
These equations assume a one-dimensional (head-on) collision. For oblique 2D collisions, the vector components must be handled separately along each axis.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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