Momentum Calculator | p=mv
Calculate momentum, impulse, and solve collision problems.
What Is the Momentum Calculator | p=mv?
The Momentum Calculator handles three core dynamics problems: single-object momentum (p = mv) with simultaneous kinetic energy output; impulse (J = FΔt) showing the change in momentum from an applied force; and 1D collision analysis for elastic, partially inelastic, and perfectly inelastic collisions with full momentum conservation verification and kinetic energy loss calculation.
- ›Multi-unit support: mass in kg, g, lb, or slug; velocity in m/s, km/h, mph, or ft/s, all converted internally.
- ›Collision solver: elastic (KE conserved), perfectly inelastic (objects stick), and partially inelastic using a coefficient of restitution (0–1).
- ›Impulse mode: calculates J = FΔt, change in momentum, and final momentum given an initial value.
- ›KE analysis: kinetic energy before and after collision, and total energy lost to heat/deformation.
Formula
| Quantity | Formula | Notes |
|---|---|---|
| Linear momentum | p = mv | kg·m/s; vector quantity |
| Kinetic energy | KE = ½mv² | Joules; scalar quantity |
| Impulse | J = FΔt = Δp | N·s = kg·m/s |
| Elastic collision v₁′ | v₁′ = (m₁−m₂)v₁ + 2m₂v₂) / (m₁+m₂) | Both p and KE conserved |
| Elastic collision v₂′ | v₂′ = (m₂−m₁)v₂ + 2m₁v₁) / (m₁+m₂) | Both p and KE conserved |
| Perfectly inelastic | v′ = (m₁v₁ + m₂v₂) / (m₁+m₂) | Objects stick together |
| Coefficient of restitution | e = (v₂′−v₁′) / (v₁−v₂) | 0 = perfectly inelastic, 1 = elastic |
How to Use
- 1Select the mode: Momentum (p = mv), Impulse (J = FΔt), or Collision.
- 2For Momentum: enter mass and velocity, choosing units from the dropdowns.
- 3For Impulse: enter force (N), time interval (s), and optionally an initial momentum.
- 4For Collision: select the collision type, enter mass and velocity for each object.
- 5For partially inelastic collisions, enter the coefficient of restitution (0 to 1).
- 6Click Calculate, or press Enter from any field.
- 7Read results in the metric cards and detailed step-by-step working below.
Example Calculation
Example 1, Momentum and KE of a car
Example 2, Elastic collision
Sign convention for velocity
Understanding Momentum | p=mv
What Is Linear Momentum?
Linear momentum p is the product of an object's mass and velocity: p = mv. It is a vector quantity, both its magnitude and direction matter. Momentum is what makes a moving object "hard to stop." A heavy truck moving slowly can have the same momentum as a small car moving quickly: doubling mass or doubling speed both double momentum.
The SI unit of momentum is kg·m/s, which is equivalent to N·s (Newton-seconds). The Law of Conservation of Momentum states that the total momentum of an isolated system (no external forces) is constant. This law holds in all collisions and explosions and is one of the most powerful tools in classical mechanics.
- ›Momentum is a vector, direction matters. Two objects moving in opposite directions can have zero total momentum.
- ›Newton's Second Law in its general form: F = dp/dt (force equals rate of change of momentum).
- ›Conservation of momentum is a consequence of Newton's Third Law: forces come in equal-opposite pairs.
- ›Momentum is conserved even in inelastic collisions where kinetic energy is not.
Impulse and the Impulse-Momentum Theorem
Impulse J is the product of a force F and the time interval Δt over which it acts: J = FΔt. The impulse-momentum theorem states that J = Δp, the impulse equals the change in momentum. This explains why sports equipment, airbags, and packaging materials work by increasing the collision time: a longer Δt for the same Δp means a smaller average force F.
- ›Airbag: increases collision time from ~10 ms to ~60 ms, reducing peak force by 6×.
- ›Cricket bat follow-through: prolongs contact time to increase impulse (and ball speed).
- ›Catching a ball: "giving way" increases Δt, reducing the force on your hands.
- ›Car crumple zones: controlled deformation increases Δt and reduces occupant force.
Types of Collisions
Collisions are classified by how much kinetic energy is conserved. All collision types conserve total momentum (in isolated systems):
- ›Elastic: both momentum and KE are conserved. Coefficient of restitution e = 1. Approximately true for billiard balls, atomic collisions.
- ›Inelastic: momentum conserved, KE is not (some lost to heat, sound, deformation). Real-world collisions are always inelastic to some degree. e < 1.
- ›Perfectly inelastic: the two objects stick together after collision. Maximum KE loss consistent with momentum conservation. e = 0.
The coefficient of restitution e = (relative speed after) / (relative speed before) = (v₂′ − v₁′) / (v₁ − v₂). A rubber ball bouncing off a hard floor has e ≈ 0.7–0.9; a lump of clay has e ≈ 0.
Momentum in Sports and Vehicle Safety
Momentum analysis underlies vehicle crash testing, sports biomechanics, and ballistics. In car crashes, the crumple zone extends the collision time so the impulse (Δp = same) is spread over longer Δt, reducing the peak force on occupants. In sports, a heavier bat gives more momentum transfer to the ball, but only if swing speed is maintained.
- ›Football tackle: conserved momentum distributes between ball carrier and defender, determining post-tackle motion.
- ›Rocket propulsion: expelling exhaust mass backward gives the rocket forward momentum (conservation in absence of gravity).
- ›Ballistics: bullet momentum = m × v; stopping power relates to momentum transfer to target.
- ›Pool/snooker: near-elastic collisions allow precise angle calculations using conservation laws.
Frequently Asked Questions
What is linear momentum and what are its units?
Linear momentum p = mv is the product of mass (kg) and velocity (m/s). Its SI unit is kg·m/s, equivalent to N·s. It is a vector quantity, direction matters.
A 1500 kg car at 20 m/s has p = 30,000 kg·m/s. A 0.145 kg baseball at 40 m/s (a fast pitch) has p = 5.8 kg·m/s, much less momentum but a much higher velocity. To stop both you need the same impulse as their respective momenta.
What is the difference between elastic and inelastic collisions?
All collision types conserve total momentum. The difference is whether kinetic energy is also conserved:
- ›Elastic: KE conserved, e = 1. Example: billiard balls, molecular collisions in ideal gases.
- ›Inelastic: KE is lost (to heat, sound, deformation), e is between 0 and 1.
- ›Perfectly inelastic: maximum KE loss, objects stick together, e = 0.
No real macroscopic collision is perfectly elastic, there is always some energy lost. Elastic collisions are idealised approximations useful in physics problems and particle physics.
What is the impulse-momentum theorem?
The impulse-momentum theorem states that the impulse J applied to an object equals its change in momentum: J = FΔt = Δp = m(v_final − v_initial).
This is the time-integrated form of Newton's Second Law (F = ma = m(Δv/Δt)). It is particularly useful when force varies with time or when you know the time duration of a collision rather than the force profile. Sports applications: a tennis racket striking a ball applies a large average force over a very short contact time (~5 ms) to produce the necessary impulse.
Is momentum conserved in all collisions?
Yes, momentum is always conserved in isolated systems (no net external force). This includes elastic, inelastic, and perfectly inelastic collisions. It also includes explosions, where one object splits into multiple pieces: total momentum before = 0 (at rest) = total momentum after (pieces fly in opposite directions with equal and opposite momenta).
Momentum is not conserved if there are external forces. A ball rolling to a stop loses momentum to friction, friction is an external force. In the collision calculator, we assume the collision duration is so short that external forces (friction, gravity) have negligible effect during the collision itself.
What is the coefficient of restitution?
The coefficient of restitution e measures how "bouncy" a collision is: e = (relative speed of separation) / (relative speed of approach) = (v₂′ − v₁′) / (v₁ − v₂).
- ›e = 1: perfectly elastic (no energy lost)
- ›e = 0: perfectly inelastic (objects stick)
- ›Rubber ball on concrete: e ≈ 0.8
- ›Steel ball bearing on steel plate: e ≈ 0.9–0.95
- ›Clay or putty: e ≈ 0
The coefficient depends on both materials and impact speed. At very high speeds, even hard materials become more inelastic due to plastic deformation.
How is momentum different from kinetic energy?
Both momentum and kinetic energy depend on mass and velocity, but they are fundamentally different quantities:
- ›Momentum p = mv is a vector (has direction), KE = ½mv² is a scalar (no direction).
- ›Momentum is conserved in all collisions; KE is only conserved in elastic collisions.
- ›Doubling velocity doubles momentum but quadruples KE.
- ›Two objects with equal momentum can have very different KEs (heavier object has less KE).
Both are calculated side-by-side in this calculator for the single-object momentum mode.
What are the supported mass and velocity units?
The calculator accepts mass in kg (SI), g (grams), lb (pounds), and slug (US customary, used in aerospace). Velocity accepts m/s (SI), km/h, mph, and ft/s. All conversions are handled automatically.
- ›1 slug = 14.5939 kg (force-based unit: 1 slug has weight of 1 lbf in standard gravity)
- ›1 lb = 0.453592 kg
- ›1 mph = 0.44704 m/s
- ›1 km/h = 1/3.6 m/s