Momentum Calculator
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Total Momentum
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Inelastic Final Velocity
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How Momentum Works
Momentum is a fundamental quantity in physics defined as the product of an object's mass and its velocity: p = mv. It is a vector quantity, meaning it has both magnitude and direction. According to Feynman's Lectures on Physics (Caltech), momentum conservation is one of the most important principles in all of physics, directly linked to the translational symmetry of space through Noether's theorem. The SI unit for momentum is kilogram meters per second (kg m/s). A 1,500 kg car traveling at 20 m/s has a momentum of 30,000 kg m/s, while a 0.145 kg baseball thrown at 40 m/s has a momentum of only 5.8 kg m/s.
The conservation of momentum states that in any closed system (no external forces), the total momentum before an interaction equals the total momentum after. According to the American Physical Society, this principle has been verified experimentally to extraordinary precision in particle physics experiments at facilities like CERN, where conservation of momentum is used to detect neutrinos and other invisible particles. This calculator computes momentum for two objects, their total system momentum, and the resulting velocity if they undergo a perfectly inelastic collision.
The Momentum Formula and Collision Equations
The core momentum equations are:
- Momentum: p = m x v (mass times velocity)
- Conservation: m1*v1 + m2*v2 = m1*v1' + m2*v2' (total momentum is constant)
- Perfectly inelastic collision: vf = (m1*v1 + m2*v2) / (m1 + m2)
- Impulse-momentum theorem: F x delta-t = m x delta-v
Worked example: A 5 kg ball moving at 10 m/s collides with a stationary 3 kg ball. If they stick together (perfectly inelastic): total initial momentum = 5(10) + 3(0) = 50 kg m/s. Final velocity = 50 / (5+3) = 6.25 m/s. Initial KE = 0.5(5)(10^2) = 250 J. Final KE = 0.5(8)(6.25^2) = 156.25 J. Energy lost = 93.75 J (37.5%), converted to heat, sound, and deformation. Use the kinetic energy calculator to verify these values.
Key Terms You Should Know
- Linear Momentum (p) — The product of mass and velocity, p = mv. A vector quantity measured in kg m/s. Doubling either mass or velocity doubles the momentum.
- Impulse (J) — The change in momentum of an object, equal to force multiplied by time: J = F x delta-t = delta-p. Measured in Newton-seconds (N s), which is dimensionally identical to kg m/s.
- Elastic Collision — A collision where both momentum and kinetic energy are conserved. Approximately realized in billiard ball collisions and ideal gas molecule interactions. Perfectly elastic collisions are theoretical; real collisions always lose some energy.
- Inelastic Collision — A collision where momentum is conserved but kinetic energy is not. Some kinetic energy is converted to thermal energy, sound, or permanent deformation. Most real-world collisions are inelastic.
- Perfectly Inelastic Collision — A special case where the colliding objects stick together after impact, resulting in maximum kinetic energy loss. The final velocity is calculated as vf = (m1*v1 + m2*v2) / (m1 + m2).
Collision Type Comparison Table
The following table compares the three types of collisions for a 5 kg object at 10 m/s striking a stationary 3 kg object. In all cases, total momentum is conserved at 50 kg m/s. According to physics research, approximately 60-80% of kinetic energy is typically lost in automobile collisions, making them highly inelastic.
| Property | Elastic | Inelastic | Perfectly Inelastic |
|---|---|---|---|
| Momentum conserved? | Yes | Yes | Yes |
| KE conserved? | Yes (250 J) | No (partial loss) | No (max loss) |
| Objects after collision | Separate | Separate | Stick together |
| v1 after (5 kg) | 2.5 m/s | Varies | 6.25 m/s (combined) |
| v2 after (3 kg) | 12.5 m/s | Varies | 6.25 m/s (combined) |
| KE after | 250 J (100%) | Variable | 156.25 J (62.5%) |
| Real-world example | Billiard balls | Tennis ball on ground | Cars that crumple together |
Practical Examples
Example 1 -- Car Collision: A 1,200 kg sedan traveling east at 15 m/s collides head-on with a 2,000 kg SUV traveling west at 10 m/s. Taking east as positive: total momentum = 1200(15) + 2000(-10) = 18,000 - 20,000 = -2,000 kg m/s (net westward). If they lock together: vf = -2,000 / 3,200 = -0.625 m/s (moving slowly westward). The initial KE was 0.5(1200)(225) + 0.5(2000)(100) = 135,000 + 100,000 = 235,000 J. Final KE = 0.5(3200)(0.625^2) = 625 J. Over 99.7% of kinetic energy was absorbed by deformation.
Example 2 -- Football Tackle: A 100 kg running back moving at 8 m/s is tackled by a 110 kg linebacker moving at -6 m/s (opposite direction). Total momentum = 100(8) + 110(-6) = 800 - 660 = 140 kg m/s. Combined velocity = 140/210 = 0.667 m/s in the running back's direction. The running back's momentum advantage means the tackle moves slightly forward, which is why heavier, faster running backs are harder to stop.
Example 3 -- Rocket Propulsion: A 10,000 kg rocket ejects 50 kg of exhaust gas at 3,000 m/s. By conservation of momentum: 0 = (10,000 - 50)(v_rocket) + 50(-3,000). Solving: v_rocket = 150,000 / 9,950 = 15.08 m/s. Each burst of exhaust incrementally increases the rocket's velocity. This principle, formalized in the Tsiolkovsky rocket equation, governs all space propulsion. Check related calculations with the projectile motion calculator.
Tips and Strategies for Momentum Problems
- Always define a positive direction before starting calculations. Momentum is a vector, so direction matters. Objects moving in opposite directions have opposite-sign velocities. Forgetting this is the most common source of errors in collision problems.
- Check your answer by verifying conservation. Total momentum before must equal total momentum after. If your calculated post-collision momenta do not sum to the initial total, there is an arithmetic error.
- Use the impulse-momentum theorem for force problems. If you know the change in momentum and the time of contact, you can find the average force: F = delta-p / delta-t. This is how crash investigators calculate impact forces and how engineers design safety equipment.
- Remember that momentum is conserved even when KE is not. In inelastic collisions, track momentum (always conserved) separately from energy (partially lost to heat, sound, deformation). Calculate the kinetic energy before and after to find energy lost.
- For 2D collisions, resolve into x and y components. Momentum is conserved independently in each direction. This makes glancing collision problems manageable by solving two separate 1D conservation equations.
- Watch units carefully. Mass in kg and velocity in m/s gives momentum in kg m/s. If using grams or km/h, convert before calculating. The unit converters page can help with these conversions.
Frequently Asked Questions
What is the law of conservation of momentum?
The law of conservation of momentum states that the total momentum of a closed system remains constant when no external forces act on it. Mathematically: m1*v1 + m2*v2 = m1*v1' + m2*v2', where primed values represent velocities after the interaction. This law holds for all types of collisions (elastic, inelastic, and perfectly inelastic) and all interactions between objects, including explosions where objects move apart. It is one of the most fundamental principles in physics, derived from Newton's Third Law, and applies at all scales from subatomic particles to galaxy collisions.
What is the difference between elastic and inelastic collisions?
In an elastic collision, both momentum and kinetic energy are conserved. Examples include billiard ball collisions and ideal gas molecule interactions. In an inelastic collision, momentum is conserved but kinetic energy is not, with some energy converted to heat, sound, or deformation. Car crashes are classic inelastic collisions. In a perfectly inelastic collision, the objects stick together after impact, resulting in maximum kinetic energy loss while still conserving momentum. The final velocity in a perfectly inelastic collision is vf = (m1*v1 + m2*v2) / (m1 + m2).
What is impulse and how does it relate to momentum?
Impulse is the product of force and the time interval over which it acts: J = F * delta-t. The impulse-momentum theorem states that impulse equals the change in momentum: J = delta-p = m * delta-v. This relationship explains why extending the time of impact reduces the force experienced. For example, airbags in cars increase the collision time from about 5 milliseconds (hitting a hard dashboard) to approximately 50-100 milliseconds, reducing the peak force on the occupant by a factor of 10-20 while producing the same total change in momentum.
How does Newton's cradle demonstrate conservation of momentum?
Newton's cradle is a device with a row of identical suspended steel balls that demonstrates both conservation of momentum and conservation of kinetic energy simultaneously. When one ball is lifted and released, it strikes the stationary row and comes to rest while the ball on the opposite end swings out with nearly the same velocity. If two balls are released, two balls swing out on the other side. This happens because the collision between steel balls is nearly perfectly elastic, so both momentum (p = mv) and kinetic energy (KE = 0.5mv^2) must be conserved, uniquely determining that the same number of balls must swing out at the same speed.
What are the SI units for momentum?
The SI unit for momentum is kilogram meters per second (kg m/s), which can also be expressed as Newton-seconds (N s) since 1 N = 1 kg m/s^2. There is no special named unit for momentum in the SI system, unlike energy (Joule) or force (Newton). In some engineering contexts, momentum is expressed in pound-seconds (lb s) in the imperial system. Angular momentum, the rotational equivalent, is measured in kg m^2/s. For subatomic particles, momentum is sometimes expressed in electron-volts divided by the speed of light (eV/c), following from Einstein's energy-momentum relation.
How is momentum used in real-world engineering and safety?
Momentum calculations are critical in automotive crash safety design, where engineers use the impulse-momentum theorem to design crumple zones, airbags, and seatbelts that extend impact time and reduce peak forces on occupants. In sports, momentum analysis helps optimize techniques in football tackles, baseball batting, and martial arts strikes. Rocket propulsion relies on conservation of momentum: expelling exhaust gas backward propels the rocket forward. Ballistics experts use momentum conservation to reconstruct collision events and bullet trajectories. In particle physics, conservation of momentum is used to detect invisible particles by measuring the momenta of all visible products.