Momentum & Collisions Calculator
Calculate momentum (p = mv), impulse and average force, and what happens when two objects collide head-on: elastic, perfectly inelastic or with any coefficient of restitution, with the kinetic energy lost.
How to Use
- Choose a calculation: momentum, mass or velocity from p = mv; impulse or average force; or a head-on collision that is elastic, perfectly inelastic or has a coefficient of restitution e.
- Pick a positive direction. Velocities the other way are negative, so a ball coming back towards you at 1 m/s is −1 m/s.
- Enter the known values with their units: kg, g or pounds; m/s, km/h or mph; newtons; seconds or milliseconds.
- For a collision, object 1 starts on the left and object 2 on the right, so object 1 must be moving faster to the right (u₁ > u₂) for them to meet.
- Read the results and the before-and-after diagram, whose arrows are drawn to scale; Show Work checks that the total momentum is unchanged.
Worked Example
A 1,500 kg car at 20 m/s. p = mv = 1,500 × 20 = 30,000 kg·m/s. Its kinetic energy is ½mv² = 300,000 J.
A 3,000 kg truck at 10 m/s hits a parked 1,000 kg car and they lock together. Momentum before: 3,000 × 10 + 1,000 × 0 = 30,000 kg·m/s. Afterwards the same momentum is shared by 4,000 kg, so v = 30,000 ÷ 4,000 = 7.5 m/s. Kinetic energy falls from 150,000 J to ½ × 4,000 × 7.5² = 112,500 J: 25% is lost to crumpling, heat and sound.
The common mistake: conserving kinetic energy when they stick. Setting ½ × 3,000 × 10² = ½ × 4,000 × v² gives v = 8.66 m/s, which is wrong. Kinetic energy is not conserved in an inelastic collision; momentum always is. The right answer is 7.5 m/s.
Show Work
Formulas
Quantity of Motion
René Descartes, in his Principles of Philosophy (1644), proposed that the total “quantity of motion” in the world, mass times speed, never changes. He ignored direction, and several of his rules of collision were wrong as a result. In 1668 the Royal Society in London asked for papers on the laws of impact, and John Wallis, Christopher Wren and Christiaan Huygens each answered; between them they showed that the conserved quantity is mass times velocity, counted with its direction.
Isaac Newton made momentum central to mechanics in the Principia (1687), where his second law says that force is the rate of change of the quantity of motion, which is the impulse equation J = FΔt = Δp. He also tested collisions of balls hung as pendulums and found that each pair separates at a fixed fraction of its approach speed, the rule now written with the coefficient of restitution e.
Momentum is conserved in every collision. Kinetic energy is conserved only in elastic ones, which is why the calculator reports both and the share of kinetic energy turned into heat, sound and deformation.
About This Calculator
This calculator solves p = mv for momentum, mass or velocity, finds the impulse of a force or the average force needed to change a velocity in a given time, and works out head-on (one-dimensional) collisions between two objects: elastic, perfectly inelastic, or with a coefficient of restitution between 0 and 1. For every collision it checks that momentum is conserved and reports the kinetic energy before and after.
Each value takes its own unit, and the before-and-after diagram draws each object’s size by its mass and each arrow by its velocity. Everything runs in your browser; nothing is sent anywhere.
Related tools: Kinetic & Potential Energy Calculator, Force Calculator, and Physics Playground.
Frequently Asked Questions
What is momentum?
Momentum is mass times velocity, p = mv, measured in kg·m/s (the same as newton-seconds). A 1,500 kg car at 20 m/s has 30,000 kg·m/s. It has a direction, and in a collision the total momentum of the objects involved stays the same.
What is impulse?
Impulse is force times the time it acts, J = FΔt, and it equals the change in momentum. Kicking a 0.43 kg ball from rest to 25 m/s in 10 ms is an impulse of 10.75 N·s, so the average force is 10.75 ÷ 0.01 = 1,075 N, about 255 times the ball’s weight.
Why do airbags and crumple zones reduce injuries?
The change in momentum is fixed, so stretching the stopping time cuts the force. Stopping a 70 kg person from 15 m/s means an impulse of 1,050 N·s: over 0.01 s that is an average of 105,000 N, but over 0.1 s it is 10,500 N, ten times less.
What is the difference between elastic and inelastic collisions?
Both conserve momentum. An elastic collision also keeps all the kinetic energy: a 1 kg ball at 2 m/s hitting an identical ball at rest stops dead and the other moves off at 2 m/s. In a perfectly inelastic collision the objects stick together and the most energy is lost, 25% when a 3,000 kg truck at 10 m/s hits a parked 1,000 kg car.
What is the coefficient of restitution?
e is the speed at which two objects separate divided by the speed at which they approached: 1 for elastic, 0 for sticking together. A ball dropped from 1 m that bounces to 0.64 m has e = √0.64 = 0.8. With e = 0.5, a 2 kg object at 3 m/s hitting a 1 kg object at −1 m/s leaves them at 1 and 3 m/s, losing 42.1% of the kinetic energy.
How do I use the Momentum & Collisions Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Accident reconstruction
A 3,000 kg truck at 10 m/s that locks onto a parked 1,000 kg car carries it along at 7.5 m/s; 37.5 kJ of the 150 kJ becomes damage and heat.
Sport
Kicking a 0.43 kg ball to 25 m/s with 10 ms of foot contact takes an average force of 1,075 N.
Safety design
Stopping 70 kg from 15 m/s in 0.1 s instead of 0.01 s cuts the average force from 105,000 N to 10,500 N.
Recoil
A 10 g bullet leaving at 400 m/s has 4 kg·m/s of momentum, so a 4 kg rifle recoils at 4 ÷ 4 = 1 m/s.
Pool and Newton’s cradle
Equal masses in a head-on elastic collision swap velocities: a ball at 2 m/s stops and the one it hits leaves at 2 m/s.
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