Math
Elastic Collision Calculator
Solve a 1D elastic or perfectly inelastic two-body collision. Get final velocities, momentum before and after, kinetic energy lost, and the coefficient of restitution.
Elastic Collision Calculator
Both momentum and kinetic energy are conserved. Bodies bounce apart with new velocities.
Final velocities (elastic)
v1' = 1 m/s, v2' = 4 m/s
Body 1 final velocity
1 m/s
Body 2 final velocity
4 m/s
Total momentum before
6 kg m/s
Total momentum after
6 kg m/s
Total KE before
9 J
Total KE after
9 J
Kinetic energy lost
0 J
Coefficient of restitution
1
Momentum conserved: yes
Kinetic energy conserved: yes
Frequently Asked Questions about the Elastic Collision Calculator
What is an elastic collision?
An elastic collision is a 1D two-body collision where both total momentum and total kinetic energy are conserved. Real-world examples are close approximations: billiard balls, atomic and subatomic particles, and steel ball bearings all bounce nearly elastically. After the collision, each body has a new velocity given by v1' = ((m1 - m2) v1 + 2 m2 v2) / (m1 + m2) and v2' = ((m2 - m1) v2 + 2 m1 v1) / (m1 + m2).
What is the difference between elastic and perfectly inelastic collisions?
Both conserve momentum. The difference is what happens to kinetic energy. In an elastic collision, KE is fully preserved, so the bodies bounce apart with the same total energy they had before. In a perfectly inelastic collision, the bodies stick together and move with a common final velocity v = (m1 v1 + m2 v2) / (m1 + m2). Some kinetic energy is converted into heat, sound, or deformation, so the system loses energy even though momentum stays the same.
What happens when two equal masses collide head-on?
If m1 equals m2 in an elastic collision, the two bodies simply exchange velocities. A moving ball striking an identical stationary ball stops dead, and the second ball moves off at the first ball's original velocity. You can verify this with the formula: when m1 = m2, the (m1 - m2) term drops out and v1' becomes v2 while v2' becomes v1.
Can the velocities be negative?
Yes. Velocity is signed because it encodes direction. Pick a positive axis (say, rightward), then bodies moving leftward have negative velocity. A head-on collision between bodies moving toward each other always has opposite signs, for example v1 = +5 and v2 = -3. Masses, however, must be strictly positive and finite. The calculator rejects zero, negative, NaN, and infinite mass inputs.
What is the coefficient of restitution?
The coefficient of restitution (e) is the ratio of the relative speed after the collision to the relative speed before: e = (v2' - v1') / (v1 - v2). It equals 1 for a perfectly elastic collision and 0 for a perfectly inelastic collision where the bodies stick together. Real collisions land in between: a basketball on hardwood has e around 0.75, a tennis ball about 0.7, and a dropped clay ball close to 0. This calculator reports the limit cases e = 1 (elastic mode) and e = 0 (inelastic mode).