Momentum Calculator

A Bullet vs a Car: Which Has More Momentum?

A 10 g bullet leaving the barrel at 900 m/s feels like the obvious winner on raw speed alone. Its momentum works out to 9 kg·m/s. A 1,500 kg car crawling at just 0.006 m/s, barely moving at all, has that exact same momentum. At a normal highway speed of 100 km/h (27.8 m/s), the car's momentum reaches 41,700 kg·m/s, more than 4,600 times the bullet's. The bullet does carry far more kinetic energy per unit mass (450,000 J/kg versus 386 J/kg for the car), because kinetic energy scales with velocity squared while momentum scales with velocity alone. That single distinction, p = mv versus KE = 0.5mv squared, is why momentum and energy tell different stories about the same collision, and it is the reason this calculator returns both figures side by side rather than just one.

How the Momentum Calculator Works

This tool finds the linear momentum of one or two objects using p = mv, where p is momentum in kilogram-metres per second, m is mass, and v is velocity. In single-object mode, enter mass and velocity in any supported unit and the calculator converts to SI before returning momentum and kinetic energy together. In two-body collision mode, enter the masses and initial velocities of two objects, pick elastic or perfectly inelastic, and the calculator applies conservation of momentum to work out post-collision velocities, momenta, and energy lost. According to the Physics Classroom momentum definition, momentum is a vector quantity, so its sign encodes direction, positive for motion one way, negative for the opposite.

Conservation of Momentum: The Core Law

The law of conservation of momentum states that total momentum in an isolated system stays constant provided no net external force acts on it. For two colliding objects: m1 x v1_initial + m2 x v2_initial = m1 x v1_final + m2 x v2_final. That single equation has two unknowns, so a second condition determines the outcome. In a perfectly inelastic collision, where the objects stick together, v1_final = v2_final = (m1v1 + m2v2) / (m1 + m2). In an elastic collision, where kinetic energy is also conserved, v1_final = ((m1 minus m2) x v1 + 2m2 x v2) / (m1 + m2) and v2_final = ((m2 minus m1) x v2 + 2m1 x v1) / (m1 + m2). The Khan Academy elastic and inelastic collision explainer walks through the full derivation using both conservation laws at once.

Momentum vs Kinetic Energy: Why They Diverge in a Real Crash

Take a 1,500 kg car hitting a stationary 1,200 kg car at 20 m/s in a perfectly inelastic collision. Momentum before: 1,500 x 20 = 30,000 kg·m/s. After: combined mass 2,700 kg, velocity 30,000 / 2,700 = 11.11 m/s, so momentum is still exactly 30,000 kg·m/s. Kinetic energy before was 0.5 x 1,500 x 400 = 300,000 J; after, 0.5 x 2,700 x 123.5 = 166,700 J. The 133,300 J difference converted into deformation, heat, and sound, which is precisely what NHTSA crashworthiness research measures when testing crumple-zone performance. Momentum was conserved exactly; kinetic energy was not, because real car crashes are inelastic rather than elastic.

Our Impulse and Momentum Calculator extends this into force-time analysis, showing how the same momentum change produces very different average forces depending on how long the contact lasts.

Accuracy and Limitations

This calculator uses exact arithmetic for conservation of momentum and kinetic energy, with every input converted to SI units before calculation so unit-conversion errors are eliminated. The equations assume a closed system with no external forces during the collision itself, a valid approximation whenever collision duration is far shorter than the time over which friction or gravity act meaningfully, which covers essentially all high-speed impacts. For a car crash lasting roughly 100 milliseconds, external forces are negligible against the collision forces, keeping the velocity outputs accurate to better than 1 percent.

The elastic model assumes zero energy loss, a theoretical ideal no macroscopic object fully achieves, though steel ball bearings in a Newton's Cradle come close at around 95 percent elastic. The perfectly inelastic model assumes the objects stay joined, while real collisions usually land somewhere between the two extremes. The HyperPhysics momentum reference covers the coefficient of restitution model for grading any degree of inelasticity in between, which falls outside what this calculator computes directly.

Reading Momentum Results Correctly

The mistake that trips people up most is forgetting momentum is a vector, so direction matters just as much as magnitude. When two objects move toward each other, their momenta point in opposite directions and must be subtracted, not added. A 1,000 kg car at +20 m/s colliding with an identical car at -15 m/s gives a total initial momentum of 1,000 x 20 + 1,000 x (-15) = 5,000 kg·m/s in the positive direction, not 35,000 kg·m/s, which is what you get by mistakenly adding the raw speeds instead of the signed momenta. Assign a positive direction before starting any calculation, then keep velocities signed consistently throughout. This calculator applies signs automatically once you enter a negative velocity, so conservation is handled correctly without any manual bookkeeping.

Momentum in Rocket Propulsion and Everyday Physics

Rockets work entirely on conservation of momentum: propellant is ejected backward at high speed, and by conservation the rocket gains equal and opposite forward momentum. Thrust force is simply the rate of momentum carried away by the exhaust, F = exhaust velocity x mass flow rate, which is why engineers chase higher exhaust velocities to squeeze more momentum change from every kilogram of propellant burned. The same underlying law governs a figure skater pulling in their arms to spin faster, a swimmer pushing off a pool wall, and the recoil of a fired rifle, all closed-system exchanges where momentum, not energy, is the quantity that has to balance.