Calculate total momentum, impulse, center-of-mass velocity, and 1D collision outcomes from mass, velocity, force, time, and restitution.
Total Momentum Formula
For a single object, momentum is the product of mass and velocity:
- p = momentum (kg·m/s, also written N·s)
- m = mass (kg)
- v = velocity (m/s)
For a system of two or more objects, total momentum is the vector sum of every individual object’s momentum:
The right-hand side shows that the total momentum equals the total system mass times the velocity of the center of mass. When the system is isolated, that center-of-mass velocity never changes, regardless of how many internal collisions or explosions occur.
What is Total Momentum?
Momentum quantifies the translational motion of an object or system. Newton’s original formulation of his second law was F = dp/dt, meaning force is the rate of change of momentum rather than mass times acceleration. That framing makes momentum the more fundamental quantity: it is what changes when a force acts, what transfers between objects during a collision, and what rockets must expel backward in order to move forward.
Because momentum is a vector, direction is not optional. Two 70 kg runners charging toward each other at 5 m/s contribute +350 kg·m/s and -350 kg·m/s to the system, for a total of zero. If they collide and hold on, the combined 140 kg mass is stationary, exactly as conservation demands. A common error is adding momentum magnitudes without signs, which gives the wrong total whenever any objects are moving in opposing directions.
Conservation of Momentum
The total momentum of an isolated system is conserved. For the classical particles modeled here, internal interaction forces cancel and only net external impulse changes system momentum. An effectively isolated collision can therefore be analyzed by momentum conservation over its contact interval. See OpenStax on linear-momentum conservation.
For a real collision, assess whether external impulse during the contact interval is small compared with the interaction impulse. The calculator assumes that it is negligible; it does not infer contact time, friction or accuracy from an object preset.
Elastic, Inelastic, and Perfectly Inelastic Collisions
Momentum is conserved for a collision system with negligible external impulse. This calculator models passive one-dimensional collisions with restitution e from 0 to 1; it excludes superelastic cases. Kinetic energy behavior depends on the collision model.
In an ideal elastic collision (e = 1), total kinetic energy is conserved. For e = 0, both bodies have the same final velocity in this model; they need not physically bond. With fixed initial momenta, this gives the minimum final translational kinetic energy. For 0 < e < 1, relative separation speed is e times the prior approach speed. A surface-impact energy ratio of e squared applies only under the corresponding idealized fixed-surface assumptions; no material-specific coefficient is guaranteed here.
Momentum Reference Values
The following illustrative masses and rounded speeds show a range of momentum scales. They are arithmetic examples, not measured values or universal specifications:
| Object | Mass | Illustrative Speed | Momentum (kg·m/s) |
|---|---|---|---|
| 9 mm bullet | 7.5 g | 370 m/s | 2.8 |
| Baseball (90 mph pitch) | 145 g | 40 m/s | 5.8 |
| NBA basketball (full-court pass) | 624 g | 12 m/s | 7.5 |
| Soccer ball (penalty kick) | 430 g | 30 m/s | 12.9 |
| NFL lineman (full sprint) | 140 kg | 4.5 m/s | 630 |
| Passenger car (60 mph) | 1,500 kg | 27 m/s | 40,500 |
| Loaded semi truck (55 mph) | 36,000 kg | 25 m/s | 900,000 |
| Boeing 747 (takeoff speed) | 300,000 kg | 83 m/s | 24,900,000 |
For the supplied table values, bullet momentum is 0.0075 × 370 = 2.775 kg·m/s and baseball momentum is 0.145 × 40 = 5.8 kg·m/s. The truck example gives 36,000 × 25 = 900,000 kg·m/s. These are momentum comparisons; stopping distance also depends on available force, friction, reaction time and other conditions.
Impulse and Momentum
The impulse-momentum theorem connects force, contact time, and momentum change:
Impulse J = average net force × elapsed time, in N·s, equals the signed change in momentum. For a fixed momentum change, increasing the force application time reduces the required average net force. Peak force depends on the time history. This calculator does not predict occupant forces, restraint performance or crash durations.
Momentum vs. Kinetic Energy
Momentum and kinetic energy both depend on mass and velocity but scale differently and play different roles in collision analysis. The relationship connecting them is KE = p²/(2m). Momentum scales linearly with velocity; kinetic energy scales with velocity squared. Doubling an object’s speed doubles its momentum but quadruples its kinetic energy. Actual crash forces and injury outcomes also depend on stopping time, deformation, geometry and other conditions; this scaling alone does not predict them.
Momentum is conserved in all isolated-system collisions. Kinetic energy is conserved only in perfectly elastic collisions. In inelastic events, the lost kinetic energy converts to heat, sound, or permanent deformation. For two equal masses sticking together with one initially stationary, half the initial kinetic energy is lost. The fraction is different for other initial velocities; equal and opposite velocities give zero final translational kinetic energy. Total momentum remains conserved when external impulse is negligible. This makes momentum the reliable bookkeeping quantity for multi-body systems, while kinetic energy requires tracking energy type conversions separately.
Fourteen tasks cover signed momentum and impulse inverses, two-body totals, restitution collisions, bullet kinetic energy and conventional power factor. Collision body 1 is assumed left of body 2 and approaching it (u1 > u2); otherwise the equations are algebraic outcomes without a contact prediction. Kinetic energy uses ½mv². Conventional power factor uses grains × feet/second ÷ 1000; displayed decimals are raw arithmetic, not an official classification or chronograph procedure. Consult USPSA’s current rules for averaging, decimal-discarding and equipment-specific requirements. Example mass presets are illustrative. The models omit relativity, external impulse, rotation and deformation details.
FAQ
What is the difference between total momentum and angular momentum?
Total linear momentum (p = mv, in kg·m/s) describes translational motion, how objects move from place to place. Angular momentum (L = Iω, in kg·m²/s) describes rotational motion. Both are independently conserved in isolated systems. A spinning figure skater pulling their arms inward increases angular velocity to conserve angular momentum, while their linear momentum remains zero because their center of mass is not translating.
Can total momentum be zero while objects are still moving?
Yes. Because momentum is a vector, opposite-direction momenta cancel. Two equal-mass objects moving toward each other at equal speeds have zero total system momentum. A stationary bomb that explodes also had zero total momentum, so all fragments must have momenta that vectorially sum to zero after detonation, which constrains the total vector momentum but does not require a spatially symmetric debris pattern.
Can total momentum be negative?
Yes. Momentum sign depends on the chosen positive direction convention. If rightward is positive, then a leftward-moving object has negative momentum. The conservation law applies to whatever sign convention you define consistently. The total can be any real number, positive, negative, or zero.
What is the difference between momentum and kinetic energy?
Momentum (p = mv) is always conserved in isolated systems and scales linearly with speed. Kinetic energy (KE = ½mv²) is conserved only in elastic collisions and scales with speed squared. The relationship between them is KE = p²/(2m). For the same momentum, a lighter object carries more kinetic energy than a heavier one, which is why high-velocity, low-mass projectiles deliver more energy per unit of momentum than slow, heavy ones.
What breaks conservation of momentum?
Any net external force applied over a finite time changes the total momentum of a system. Common examples include friction from a surface outside the defined system, gravity when the gravitational source is not included in the system, and normal forces from external walls or constraints. Expanding the system boundary to include the external body usually internalizes the force and restores momentum conservation for that larger system.
