Enter the total velocity and the total sum of the masses into the calculator to determine the Total Momentum. Use the tabs to switch between basic momentum, impulse, two-body systems, and 1D collision analysis.
Total Momentum Formula
For a single object, momentum is the product of mass and velocity:
p = m \cdot v
- 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:
p_{total} = m_1 v_1 + m_2 v_2 + \cdots + m_n v_n = M_{total} \cdot v_{cm}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 never changes. This is an exact result, not an approximation, and follows directly from Newton’s third law. Every internal force between two parts of a system is matched by an equal and opposite reaction force, so all internal forces sum to exactly zero and cannot alter total momentum. Only a net external force from outside the defined system boundary can do that.
In real collision problems, perfect isolation is an approximation that is usually excellent. A billiard ball collision lasts roughly 0.5 ms, during which table friction produces a momentum change thousands of times smaller than the collision forces themselves. Treating the two-ball system as isolated introduces negligible error over that window. This is the impulse approximation: when collision forces dominate all other forces over the brief contact duration, momentum conservation holds to engineering accuracy.
Elastic, Inelastic, and Perfectly Inelastic Collisions
Momentum is conserved in every collision type. Kinetic energy behaves differently depending on the collision, characterized by the coefficient of restitution (e), which ranges from 0 to 1.
In an elastic collision (e = 1), both momentum and kinetic energy are fully conserved. No kinetic energy converts to heat or deformation. Hard steel ball bearings and atomic-scale gas molecule collisions approximate this condition well. In a perfectly inelastic collision (e = 0), the objects merge into one and move together. This is the maximum possible kinetic energy loss while still satisfying momentum conservation. In a partially inelastic collision (0 < e < 1), objects separate after contact but with reduced relative velocity. A basketball dropped onto hardwood has e approximately 0.85 and retains about 72% of its kinetic energy per bounce, since the kinetic energy ratio equals e squared.
Momentum Reference Values
Momentum spans roughly ten orders of magnitude across objects encountered in daily life. These values provide useful scale when interpreting calculator outputs:
| Object | Mass | Typical 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 |
A notable pattern from the table: the 9 mm bullet carries less momentum than a pitched baseball despite traveling 9 times faster. The baseball’s mass advantage (about 19 times heavier) more than compensates. A loaded semi truck carries roughly 155,000 times the momentum of a baseball, almost entirely due to mass rather than speed. This asymmetry is why stopping large vehicles requires extended braking distances even at modest highway speeds.
Impulse and Momentum
The impulse-momentum theorem connects force, contact time, and momentum change:
J = \Delta p = F_{avg} \cdot \Delta tImpulse (J) is the product of average force and the time over which it acts. Its units are newton-seconds (N·s), which are dimensionally identical to kg·m/s. The theorem states that the total momentum change of an object equals the net impulse applied to it. This has direct engineering consequences: an airbag extends a crash collision duration from roughly 10 ms to 30 to 40 ms, cutting the average force on an occupant by a factor of 3 to 4 while delivering the exact same total momentum change (vehicle goes from 60 mph to zero in both cases). Crumple zones, padded interiors, and spring-loaded bumpers operate on this same principle of trading peak force for duration.
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. This scaling difference is why highway crash severity grows so much faster than braking force requirements as speed increases.
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 a perfectly inelastic collision between two equal masses, exactly half of the original kinetic energy is lost, while all the momentum is preserved in the merged object. This makes momentum the reliable bookkeeping quantity for multi-body systems, while kinetic energy requires tracking energy type conversions separately.
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 is why debris fans outward symmetrically from a stationary explosion.
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.
