Momentum Calculator

About Momentum Calculator

The Momentum Formula

Linear momentum equals mass times velocity: p = mv. It is a vector quantity, meaning direction matters. The SI unit is kilogram-metres per second (kg m/s). Momentum is conserved in every closed system - this is one of the most fundamental principles in physics.

Solve For	Formula	Example
Momentum	p = m x v	5 kg x 10 m/s = 50 kg m/s
Mass	m = p / v	50 kg m/s / 10 m/s = 5 kg
Velocity	v = p / m	50 kg m/s / 5 kg = 10 m/s

Worked example: a 0.145 kg baseball thrown at 40 m/s has momentum p = 0.145 x 40 = 5.8 kg m/s. A 70 kg person walking at 1.4 m/s has momentum p = 70 x 1.4 = 98 kg m/s. The person has 17 times more momentum despite moving 29 times slower, because mass is 483 times greater.

Impulse and the Impulse-Momentum Theorem

Impulse is the change in momentum caused by a force acting over a time interval: J = F x delta-t = delta-p. This explains why airbags save lives - they increase the time of deceleration, reducing the peak force on the body for the same change in momentum.

Scenario	Mass	Speed Change	Time	Average Force
Catching a baseball (soft hands)	0.145 kg	40 m/s to 0	0.1 s	58 N
Catching a baseball (stiff hands)	0.145 kg	40 m/s to 0	0.01 s	580 N
Car crash without airbag	70 kg	13.9 m/s to 0	0.05 s	19,460 N
Car crash with airbag	70 kg	13.9 m/s to 0	0.3 s	3,243 N

The airbag reduces the stopping force by 6x simply by extending the collision time from 0.05 to 0.3 seconds. Same change in momentum, much lower peak force.

Elastic vs Inelastic Collisions

In all collisions, total momentum is conserved. The difference is what happens to kinetic energy.

Type	Momentum	Kinetic Energy	Objects After	Example
Perfectly elastic	Conserved	Conserved	Bounce apart	Billiard balls, atomic collisions
Inelastic	Conserved	Partially lost	Bounce apart (deformed)	Most real-world crashes
Perfectly inelastic	Conserved	Maximum loss	Stick together	Clay balls, football tackle

Collision Formula Reference

For elastic collisions between two objects, the final velocities are:

Variable	Formula
v1_final	((m1 - m2) x v1 + 2 x m2 x v2) / (m1 + m2)
v2_final	((m2 - m1) x v2 + 2 x m1 x v1) / (m1 + m2)

For perfectly inelastic collisions (objects stick together):

Variable	Formula
v_final	(m1 x v1 + m2 x v2) / (m1 + m2)
KE lost	KE_initial - KE_final

Worked example: a 2 kg ball moving at 5 m/s hits a stationary 3 kg ball in a perfectly inelastic collision. Final velocity = (2 x 5 + 3 x 0) / (2 + 3) = 10/5 = 2 m/s. Initial KE = 1/2 x 2 x 25 = 25 J. Final KE = 1/2 x 5 x 4 = 10 J. Energy lost = 15 J (60%), converted to heat and deformation.

Real-World Momentum Values

Object	Mass	Velocity	Momentum (kg m/s)
Ping pong ball (serve)	2.7 g	30 m/s	0.081
Cricket ball (fast bowl)	0.163 kg	42 m/s (150 km/h)	6.8
Person running	70 kg	8 m/s	560
Car on motorway	1,500 kg	31 m/s (112 km/h)	46,500
Freight train	4,000,000 kg	22 m/s (80 km/h)	88,000,000

A freight train at 80 km/h has about 1,900 times more momentum than a car at 112 km/h. This is why trains need several kilometres to stop and why level crossing safety is critical.

Why Momentum Matters for Road Safety

Momentum scales linearly with mass but also with velocity, so doubling a vehicle's speed doubles its momentum and the force required to stop it in the same time. The UK Department for Transport reported 1,602 road fatalities in Great Britain in 2024, with car occupants making up 692 deaths and vulnerable road users (pedestrians, motorcyclists, cyclists) accounting for about half of all fatalities. In the United States, NHTSA estimated 39,345 traffic fatalities in 2024, the first year below 40,000 since 2020. Momentum analysis is central to crash reconstruction: investigators back-calculate pre-crash speeds from post-crash trajectories using conservation of momentum, then use impulse-momentum to estimate the forces applied to occupants.

Seat belts, crumple zones, and airbags all exploit the same physics principle: extend delta-t to reduce the peak force for the same change in momentum. A typical modern crumple zone deforms over roughly 0.5 to 0.7 metres during a frontal impact, extending the effective stopping time from the fraction of a second seen in a rigid collision to several hundred milliseconds.

How Is Momentum Different from Kinetic Energy?

Momentum (p = mv) is linear in velocity and a vector, while kinetic energy (KE = 1/2 mv²) is quadratic in velocity and a scalar. A bullet and a lorry can carry the same momentum at very different kinetic energies, and a cricket ball and a bowling ball at the same speed have different momenta but also very different KE.

Quantity	Formula	Units	Vector?	Conserved In
Momentum	p = mv	kg m/s	Yes	All closed-system collisions
Kinetic energy	KE = 1/2 mv²	J (kg m²/s²)	No	Elastic collisions only
Impulse	J = F x delta-t	N s (kg m/s)	Yes	Equals change in momentum

This distinction matters in practice. Doubling speed doubles momentum but quadruples kinetic energy. That is why stopping distance (dominated by energy dissipation into the brake discs and tyres) grows with the square of speed, while the impulse a barrier has to absorb grows linearly.

Conservation of Momentum in Explosions and Recoil

Conservation of momentum applies to more than collisions. Any closed system where internal forces redistribute momentum, such as a gun recoiling when a bullet leaves the barrel, a rocket expelling exhaust gas, or a swimmer pushing off from a wall, follows the same rule: the vector sum of all momenta stays constant. If a 4 kg rifle fires a 0.010 kg bullet at 400 m/s, the rifle recoils at v = (0.010 x 400) / 4 = 1 m/s in the opposite direction. This simple balance is the foundation of rocket propulsion: a rocket gains forward momentum equal in magnitude to the backward momentum of the exhaust, with the specific impulse metric expressing how efficiently a propellant converts mass into thrust.

In a two-stage firework, the casing explodes mid-flight into fragments travelling in many directions. The combined momentum of every fragment must still equal the casing's momentum at the moment of explosion. Forensic ballistics teams use this principle to back-calculate the original trajectory of a projectile that broke apart.

Coefficient of Restitution and Real Collisions

Most real collisions are neither perfectly elastic nor perfectly inelastic. The coefficient of restitution (e) measures how elastic a collision is: e = 1 for perfectly elastic, e = 0 for perfectly inelastic, and everything in between for real materials. It is defined as the ratio of relative separation velocity to relative approach velocity: e = -(v1f - v2f) / (v1i - v2i).

Collision Type	Coefficient of Restitution (e)	Notes
Superball on concrete	0.85-0.90	Near elastic, bounces to ~75% of drop height
Tennis ball on racquet	0.70-0.75	Regulated by ITF to 0.53-0.58 on concrete
Basketball on wood floor	0.76	NBA requires 1.27-1.37 m bounce from 1.83 m drop
Golf ball on driver	0.83	USGA caps at 0.83 (the "COR limit")
Clay lump on concrete	~0.05	Near perfectly inelastic
Two cars in a fender bender	0.10-0.20	Crumple zones absorb most of the KE

When COR is between 0 and 1, total momentum is conserved but kinetic energy loss can be calculated as KE_lost = 1/2 x (m1 x m2 / (m1 + m2)) x (v1 - v2)² x (1 - e²). Sports equipment regulations often cap COR to keep the game challenging: a driver that bounces the ball too efficiently would push drive distances far beyond what courses are designed for.

Tips for Solving Momentum Problems

Use negative velocities for objects moving in opposite directions - this automatically handles the vector nature of momentum. In a head-on collision, if object A moves at +5 m/s and object B at -3 m/s, their momenta partially cancel. Always check your answer by verifying that total momentum before equals total momentum after the collision.

Common mistakes to avoid: mixing units (grams with kilograms, or km/h with m/s - always convert to SI before calculating), forgetting that velocity is a vector so opposing directions need opposite signs, assuming kinetic energy is conserved in every collision (only perfectly elastic collisions conserve KE), and using weight in newtons or pounds-force where mass in kilograms or pounds-mass is required. For rotational problems, angular momentum L = I x omega is the rotational analogue and follows the same conservation rules.

For energy analysis of collisions, the kinetic energy calculator computes KE = 1/2 mv² for individual objects. For finding the forces involved, the force calculator applies Newton's Second Law. For velocity and acceleration problems leading up to the collision, the velocity calculator solves kinematics equations. All calculations run in your browser with no data sent anywhere.

Sources

• GOV.UK - Reported Road Casualties Great Britain, Annual Report 2024
• NHTSA - 2024 Traffic Fatality Estimate
• The Physics Hypertextbook - Linear Momentum
• OpenStax University Physics - Types of Collisions
• NIST - SI Units: Mass
• HyperPhysics (Georgia State) - Elastic and Inelastic Collisions