Conservation Of Momentum Calculator

Calculate the final velocities of two colliding objects and confirm that total momentum is conserved, using m1v1i + m2v2i = m1v1f + m2v2f for perfectly elastic or perfectly inelastic collisions.

Results

Calculated
Total momentum
—
p = m1v1i + m2v2i, conserved throughout
Final velocity, Object 1
—
v1f after the collision
Final velocity, Object 2
—
v2f after the collision
Momentum after (check)
—
m1v1f + m2v2f, should match total momentum

Ready

Enter both masses and initial velocities, choose a collision type, then press Calculate.

How to use this calculator

This tool applies the law of conservation of momentum to a two-body collision. Enter the mass and initial velocity of each object, choose whether the collision is perfectly elastic or perfectly inelastic, and press Calculate to get each object's final velocity plus the total momentum before and after — which should match. Click Clear to reset all fields to the example values.

The formula

Conservation of momentum states that for an isolated system (no external forces), total momentum is the same before and after an interaction:

m1·v1i + m2·v2i = m1·v1f + m2·v2f

Here m1 and m2 are the two masses, v1i and v2i are their velocities before the collision, and v1f and v2f are their velocities afterward. Momentum (p = m·v) is measured in kilogram-meters per second (kg·m/s).

Elastic vs. perfectly inelastic collisions

The equation above only tells you that total momentum is conserved — it takes a second condition to solve for both unknown final velocities. This calculator supports the two standard idealized cases:

  • Perfectly elastic: kinetic energy is also conserved, so the objects separate after impact with v1f = ((m1−m2)/(m1+m2))·v1i + (2m2/(m1+m2))·v2i and v2f = (2m1/(m1+m2))·v1i + ((m2−m1)/(m1+m2))·v2i.
  • Perfectly inelastic: the objects stick together and share one common final velocity, v_f = (m1·v1i + m2·v2i) / (m1+m2). Kinetic energy is lost to heat, sound, and deformation, but momentum is unaffected.

Understanding the inputs and sign convention

Pick one direction as positive (for example, rightward or "forward") and enter velocities in that same direction consistently; motion the opposite way is a negative number. Masses must be positive and both objects' velocities should use the same unit (m/s). Mixing sign conventions or units between the two objects will produce a meaningless result.

Interpreting the results

The total momentum card shows p = m1v1i + m2v2i, the conserved quantity for this system. The two final-velocity cards show how fast each object moves after the collision — under perfectly inelastic conditions these two values will be identical, since the objects move together. The "momentum after" card recomputes m1v1f + m2v2f as a check: it should equal the total momentum card (aside from rounding), confirming the calculation is internally consistent.

Frequently Asked Questions

What is the law of conservation of momentum?
For an isolated system with no external forces, total momentum before an interaction equals total momentum after it: m1v1i + m2v2i = m1v1f + m2v2f. This holds for every collision and explosion, whether or not kinetic energy is also conserved.
What is the difference between elastic and perfectly inelastic collisions?
In a perfectly elastic collision both momentum and kinetic energy are conserved, so the objects bounce apart with separate final velocities given by v1f = ((m1−m2)/(m1+m2))v1i + (2m2/(m1+m2))v2i and v2f = (2m1/(m1+m2))v1i + ((m2−m1)/(m1+m2))v2i. In a perfectly inelastic collision the objects stick together and move at one common final velocity, v_f = (m1v1i + m2v2i)/(m1+m2); momentum is still conserved but kinetic energy is lost to heat and deformation.
Why does a final velocity come out negative?
Momentum and velocity are vector quantities, so a sign convention is required: pick one direction as positive and treat motion the opposite way as negative. A negative final velocity simply means that object ends up moving in the negative direction, not that anything is wrong with the calculation.
Does conservation of momentum apply to explosions and recoil, not just collisions?
Yes. The same equation governs any isolated interaction, including explosions and recoil, where objects start together and fly apart. Total momentum before the split (often zero if everything starts at rest) must equal the vector sum of momenta afterward, which is why a rifle recoils backward when a bullet fires forward.