Momentum Calculator
Calculate momentum p = mv, impulse J = FΔt, and the final velocities of a 1D collision from a coefficient of restitution. Signed convention and kinetic energy lost shown.
Result
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Kinetic energy lost
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p = m v; J = F Δt = Δp; F = Δp / Δt; v1, v2 from e = (v2 - v1) / (u1 - u2)
How?
How this is calculated
Momentum solves the single relationship p = m v for whichever quantity you leave blank. Impulse uses the impulse-momentum theorem J = F Δt, which equals the change in momentum Δp = m (v_f - v_i); the net-force mode inverts it to F = Δp / Δt, the single net force implied by a momentum change over a time interval.
The collision mode is a general one-dimensional restitution solver keyed on a single coefficient of restitution e in the range 0 to 1, where e = 1 is a perfectly elastic collision (kinetic energy conserved) and e = 0 is perfectly inelastic (the bodies move together). The final velocities come from v1 = (m1 u1 + m2 u2 - m2 e (u1 - u2)) / (m1 + m2) and v2 = (m1 u1 + m2 u2 + m1 e (u1 - u2)) / (m1 + m2), and the restitution check recomputes e = (v2 - v1) / (u1 - u2). Values display to 4 significant figures; the calculation runs at full precision in SI units.
Formula: p = m v; J = F Δt = Δp; F = Δp / Δt; v1, v2 from e = (v2 - v1) / (u1 - u2)
How to read a negative answer
Every velocity and momentum here is a signed one-dimensional scalar. Send a 3 kg ball at 4 m/s to the right into a 5 kg ball closing at 6 m/s from the right (u2 = -6), and the elastic solve returns v1 = -8.5 m/s: the first ball rebounds and now moves left. The sign is the direction, read against the positive axis you chose.
Elastic, inelastic, and everything between
The coefficient of restitution is one dial from a perfect bounce to a perfect stick. At e = 1 the collision is elastic and kinetic energy is conserved. At e = 0 the bodies leave with one shared velocity, the center-of-mass velocity, and the tool marks them as moving together. Any value between the two is partially inelastic, distinct velocities with some energy lost.
Where the kinetic energy goes
Momentum is conserved in every collision on this page. Kinetic energy is not: for any e below 1, some of it converts to heat, sound, and deformation. The tool reports the kinetic energy lost as a first-class number so you can see exactly how much left the mechanical system. For the energy that stays and does work, see the work, energy, and power calculator.
Assumptions and conventions
The system is isolated and one-dimensional, with no external impulse acting during the collision. Full two-dimensional vector momentum belongs to the projectile and vector lane. Mass is strictly positive; the tool flags a mass of zero or below rather than returning a meaningless answer. The unit system is SI internally, and US customary input is offered with mass in pounds and force in pound-force converted through stated SI factors, so the pound-mass and pound-force ambiguity never reaches the calculation.
Sources
- OpenStax Physics, 8.1 Linear Momentum, Force, and Impulse. OpenStax. Retrieved .
- OpenStax Physics, 8.2 Conservation of Momentum. OpenStax. Retrieved .
- OpenStax Physics, 8.3 Elastic and Inelastic Collisions. OpenStax. Retrieved .
- Wikipedia: Coefficient of restitution. Wikipedia. Retrieved .