Momentum Calculator — Collisions & p = mv

Calculate momentum (p = mv) plus perfectly inelastic and perfectly elastic collision outcomes, with kinetic energy checked before and after.

July 14, 2026

90% found this helpful

Reference Values

Last verified: Jul 14, 2026

Category	Range	What It Means	Status
Basic momentum	p = m × v	Momentum equals mass (kg) times velocity (m/s). Result is in kilogram-meters per second (kg·m/s), a vector quantity that points in the same direction as the velocity.	Good
Perfectly inelastic collision	v' = (m1v1 + m2v2) / (m1 + m2)	Objects collide and stick together, moving with one shared final velocity. Total momentum is conserved but kinetic energy is not — some is lost to heat, sound, and deformation.	Good
Perfectly elastic collision — object 1 ★	v1' = ((m1 − m2)v1 + 2m2v2) / (m1 + m2)	Closed-form solution for object 1's final velocity when both momentum and kinetic energy are conserved (objects bounce apart without any energy loss).	★ Best
Perfectly elastic collision — object 2 ★	v2' = ((m2 − m1)v2 + 2m1v1) / (m1 + m2)	Closed-form solution for object 2's final velocity in the same perfectly elastic collision. Used together with the object 1 formula above.	★ Best
Equal-mass elastic collision ★	Velocities exchange completely	Special case: when m1 = m2 in a perfectly elastic collision, object 1 takes on object 2's original velocity and vice versa — the classic billiard-ball result.	★ Best
General (partially elastic) collisions	Requires coefficient of restitution (e)	Real-world collisions are rarely perfectly elastic or perfectly inelastic. Modeling anything in between requires a coefficient of restitution (0 < e < 1), which is outside this calculator's scope — only the two textbook limiting cases are covered here.	Poor

Source: OpenStax University Physics Volume 1, Chapter 8.3 'Elastic and Inelastic Collisions' (openstax.org); Physics LibreTexts, 4.7 'Totally Elastic Collisions' (phys.libretexts.org). Both are standard closed-form textbook derivations from conservation of momentum and conservation of kinetic energy.

Worked Examples

Basic Momentum — Car on a Highway

Mass: 1,200 kg
Velocity: 25 m/s

30,000 kg·m/s

p = m × v = 1,200 × 25 = 30,000 kg·m/s.

Basic Momentum — Bowling Ball

Mass: 7 kg
Velocity: 8 m/s

56 kg·m/s

p = m × v = 7 × 8 = 56 kg·m/s — much smaller than the car despite similar speed order of magnitude, because mass is far smaller.

Perfectly Inelastic Collision — Train Cars Coupling

Mass 1: 1,000 kg
Velocity 1: 20 m/s
Mass 2: 1,500 kg
Velocity 2: 0 m/s (stationary)

8 m/s combined velocity

v' = (m1v1 + m2v2) / (m1+m2) = (1,000×20 + 1,500×0) / 2,500 = 20,000 / 2,500 = 8 m/s.

Perfectly Elastic Collision — Equal-Mass Billiard Balls

Mass 1: 1 kg
Velocity 1: 4 m/s
Mass 2: 1 kg
Velocity 2: 0 m/s (stationary)

Object 1 stops (0 m/s), Object 2 moves at 4 m/s

v1' = ((1−1)×4 + 2×1×0)/(1+1) = 0 m/s. v2' = ((1−1)×0 + 2×1×4)/(1+1) = 4 m/s. Equal masses fully exchange velocities — the classic billiard-ball result.

Perfectly Elastic Collision — Unequal Masses

Mass 1: 3 kg
Velocity 1: 4 m/s
Mass 2: 1 kg
Velocity 2: 0 m/s (stationary)

Object 1 slows to 2 m/s, Object 2 speeds off at 6 m/s

v1' = ((3−1)×4 + 2×1×0)/(3+1) = 8/4 = 2 m/s. v2' = ((1−3)×0 + 2×3×4)/(3+1) = 24/4 = 6 m/s. Check: momentum before = 3×4=12, after = 3×2+1×6=12 ✓. Kinetic energy before = 0.5×3×4²=24 J, after = 0.5×3×2²+0.5×1×6²=6+18=24 J ✓ — both conserved, confirming a perfectly elastic collision.

How to Use This Calculator

1

Pick a mode

Basic Momentum for a single object's p = mv, Inelastic Collision for two objects that stick together, or Elastic Collision for two objects that bounce apart with no energy loss.
2

Enter mass and velocity

Mass in kilograms (always positive), velocity in meters per second (can be negative for objects moving in the opposite direction).
3

For collisions, enter both objects

Mass and velocity for object 1 and object 2. A stationary object simply has a velocity of 0.
4

Read the result

Basic mode shows momentum in kg·m/s. Collision modes show the final velocity (or velocities) plus kinetic energy before and after, so you can see how much energy the collision lost — or confirm it lost none at all.

What Each Value Means

Momentum (p) (kg·m/s): The product of an object's mass and velocity — a measure of how hard it would be to stop the object. A heavy, slow object and a light, fast object can have the same momentum.
Perfectly inelastic collision (m/s (final velocity)): A collision where the two objects stick together after impact and move with one shared final velocity. Momentum is conserved but kinetic energy is not.
Perfectly elastic collision (m/s (final velocities)): A collision where the two objects bounce apart and both momentum and kinetic energy are fully conserved — no energy is lost to heat, sound, or deformation.
Kinetic energy (KE) (joules (J)): The energy an object has because of its motion, calculated as ½mv². Shown before and after each collision so you can check how much (if any) was lost.

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Frequently Asked Questions

What's the difference between an elastic and inelastic collision?

Both conserve total momentum — that's true of every collision. The difference is kinetic energy. In a perfectly elastic collision, kinetic energy is also conserved: the objects bounce apart and the total KE before equals the total KE after. In a perfectly inelastic collision, the objects stick together and move with one shared final velocity, and some kinetic energy is always lost — converted to heat, sound, and permanent deformation. Most real-world collisions (car crashes, dropped balls) fall somewhere between these two textbook extremes.

Why does this calculator only cover the two extreme cases?

Perfectly elastic and perfectly inelastic collisions both have clean closed-form formulas you can solve directly from conservation of momentum (and, for the elastic case, conservation of kinetic energy too). Everything in between — a basketball bouncing with some energy loss, two cars crumpling but not fully sticking together — requires a coefficient of restitution (e), a measured value between 0 and 1 that depends on the specific materials involved. There's no universal formula for it, so modeling a partially elastic collision accurately requires a real-world measurement this calculator can't provide.

Is momentum always conserved, even when kinetic energy isn't?

Yes, as long as no outside force acts on the system during the collision (this is called a closed or isolated system). Momentum conservation comes directly from Newton's third law — whatever force object 1 exerts on object 2 during the collision, object 2 exerts an equal and opposite force back on object 1, for exactly the same amount of time. Those forces cancel out for the system as a whole, so total momentum before always equals total momentum after, whether the collision is elastic, inelastic, or anywhere in between.

What happens when two equal masses collide elastically?

They completely exchange velocities — this is the classic billiard-ball result. If a moving ball strikes an identical stationary ball head-on, the moving ball stops dead and the stationary ball takes off with the exact velocity the first ball had. You can verify this directly from the elastic collision formulas: when m1 = m2, the (m1−m2) term in the v1' formula becomes zero, leaving v1' = v2 (object 1 takes object 2's velocity) and v2' = v1 (object 2 takes object 1's velocity).

Can momentum be negative?

Yes. Momentum is a vector, meaning it has both size and direction, and in one-dimensional problems like the ones this calculator solves, direction is represented by the sign of the velocity. If you define rightward motion as positive, an object moving left has negative velocity and therefore negative momentum. This matters most in collision problems where two objects move toward each other — enter one velocity as positive and the other as negative to represent them correctly, or your combined-velocity result will be wrong.