Conservation Of Momentum Calculator - Elastic & Inelastic 1D

A conservation of momentum calculator is a physics tool that solves one-dimensional two-body collisions by tracking how linear momentum and kinetic energy change between the initial and final states. It applies the conservation of momentum formula p1i + p2i = p1f + p2f to a user-entered pair of masses and velocities, so you can check collisions, work homework, or sanity-check a numerical simulation. Use it in physics, sports engineering, vehicle safety, robotics, and any setting where two objects meet along a straight line.

• • Physics homework: Solve a 1D elastic or inelastic problem and verify the momentum-total rows match before and after.
• • Lab and simulation checks: Confirm a measured velocity pair agrees with momentum conservation within tolerance.
• • Sports and vehicle design: Estimate post-impact speeds for two-body crashes where a coefficient of restitution is reasonable.
• • Robotics and games: Pick final velocities for billiards, hockey pucks, or simple character collisions in 2D engines that pre-compute 1D behavior.

Because the law is universal for any isolated two-body system, the same calculator handles a 2 kg cart on an air track or a 2 tonne vehicle in a low-speed bump, as long as you stay in 1D and treat external forces as negligible during the short impact. The page also switches between elastic, perfectly inelastic, and partially inelastic cases with a custom coefficient of restitution, so you can move from ideal physics into more realistic engineering estimates without leaving the same form.

Before working through a 1D collision, it helps to review how force, mass, and acceleration interact using the Forces and Newton's Laws Calculator, since momentum is the time integral of the net external force.

The calculator keeps linear momentum constant for an isolated two-body system, then layers a second condition (equal final kinetic energy for elastic collisions, a shared final velocity for perfectly inelastic collisions, or a user-set coefficient of restitution) to pin down the final velocities. Mass and velocity inputs are converted to SI base units, the right linear system is solved, and the totals are returned for verification.

• p1i: Initial momentum of object 1, equal to m1 times its initial velocity v1.
• p2i: Initial momentum of object 2, equal to m2 times its initial velocity v2.
• p1f: Final momentum of object 1, equal to m1 times its final velocity v1f.
• p2f: Final momentum of object 2, equal to m2 times its final velocity v2f.
• e: Coefficient of restitution, the ratio of relative speed after to before. e = 1 is elastic, e = 0 is perfectly inelastic.

The momentum row in the results panel is the most important verification: a correct entry keeps total momentum equal before and after, while kinetic energy may differ by the amount converted to heat, sound, or deformation.

According to OpenStax University Physics Volume 1, Section 9.2, the total linear momentum of an isolated two-body system is conserved in any collision, and the specific 1D form p1_initial + p2_initial = p1_final + p2_final holds whether the collision is elastic, inelastic, or perfectly inelastic.

Because the kinetic-energy rows in this calculator come from 1/2 m v^2, you can cross-check each energy value with the Kinetic Energy Calculator when you want the energy in kilojoules or foot-pounds.

Each of these concepts plays a different role in the conservation of momentum formula, so it helps to keep them straight before reading a final-velocity answer.

Sign matters in a 1D collision, and the Velocity Calculator helps confirm direction before trusting the momentum totals.

The form is laid out so a 1D collision can be entered in roughly the same order it would be drawn on a free-body diagram: mode, mass and velocity of object 1, then mass and velocity of object 2.

1. 1 Choose the collision mode: Pick elastic, perfectly inelastic, or partially inelastic with a custom coefficient of restitution.
2. 2 Enter m1 and v1: Type the mass in kilograms and the initial velocity in meters per second of the first object.
3. 3 Enter m2 and v2: Type the mass and velocity of the second object. Use 0 if it starts at rest.
4. 4 Set the coefficient of restitution: For partially inelastic mode, set e between 0 and 1. Leave at 1 for elastic work.
5. 5 Read the final velocities and totals: The result panel returns v1f, v2f, the total momentum before and after, and the total kinetic energy before and after.
6. 6 Verify the momentum totals match: A correct entry keeps the total momentum row equal before and after. If the numbers disagree, re-check the signs of the input velocities.

Worked example: a 0.145 kg baseball moving at 40 m/s is hit back at 50 m/s. In elastic mode with m1 = 0.145, v1 = 40, m2 = 0.145, v2 = -50, the calculator returns v1f = -50 m/s and v2f = 40 m/s, with both momentum and kinetic energy totals matching before and after.

Once you have the post-collision velocity, the Projectile Motion Calculator can follow each object through the air if the collision happens at the edge of a track.

These benefits come up in real classroom, lab, and engineering workflows where momentum conservation saves time that would otherwise be spent on algebra.

• • Skip the algebra: Both the elastic closed form and the perfectly inelastic shared-velocity formula are entered in one click.
• • See conservation directly: The total-momentum-before and total-momentum-after rows let you verify p_total stays constant in every mode.
• • Compare energy loss across modes: The kinetic-energy-before and kinetic-energy-after rows make it obvious that elastic collisions keep all kinetic energy while perfectly inelastic collisions lose the maximum amount.
• • Test edge cases safely: Equal masses, equal and opposite velocities, both objects at rest, and out-of-range restitution values are all flagged with validation messages.

The kinetic-energy rows in this calculator pair naturally with the work-energy picture developed in the Work Energy Power Calculator, which is the quickest way to see where the lost energy in an inelastic collision goes.

Several practical factors decide whether a measured or modeled collision really does conserve momentum, and whether the energy rows in this calculator match what you see in the lab.

• • This calculator covers 1D two-body collisions only. Glancing and multi-body collisions need 2D momentum components or repeated pairwise solves.
• • The model uses classical (Newtonian) momentum. At speeds approaching light speed, relativistic momentum p = gamma m v is required.
• • The kinetic-energy rows are only conserved for elastic collisions. A non-zero delta in the kinetic-energy totals is a feature of the physics, not a calculator bug.

According to OpenStax University Physics Volume 1, Section 9.4, the closed-form equations for a 1D elastic collision are v1f = ((m1 - m2) * v1 + 2 m2 v2) / (m1 + m2) and v2f = ((m2 - m1) * v2 + 2 m1 v1) / (m1 + m2), and a perfectly inelastic collision ends with a shared final velocity v_f = (m1 * v1 + m2 * v2) / (m1 + m2).

According to Lumen Learning SUNY OER University Physics, Chapter 9.4, in a perfectly inelastic collision the two objects stick together and move with the shared velocity v_f = (m1 * v1 + m2 * v2) / (m1 + m2), which conserves linear momentum but not kinetic energy.

If the post-collision trajectory bends into a curve, the Centripetal Force Calculator gives the centripetal force for the new circular path.