Conservation of Momentum Calculator
Enter the mass and initial velocity of two objects to find their final velocities and see how momentum and kinetic energy change during the collision. Choose elastic (kinetic energy conserved), perfectly inelastic (objects stick together), or explosion (one body splits into two). The calculator shows a step-by-step breakdown of the math and a momentum comparison chart.
Formula
Worked example
A 2 kg ball moving at 3 m/s collides elastically with a 1 kg ball moving at -1 m/s. Total momentum before = 2*3 + 1*(-1) = 5 kg*m/s. After: v1 = [(2-1)*3 + 2*1*(-1)] / 3 = 1/3 m/s; v2 = [(1-2)*(-1) + 2*2*3] / 3 = 13/3 m/s. Momentum after = 2*(1/3) + 1*(13/3) = 5 kg*m/s - verified.
What is conservation of momentum?
The law of conservation of momentum states that in a closed, isolated system, the total momentum before an event equals the total momentum after it. Momentum is a vector quantity defined as mass multiplied by velocity (p = mv), so direction matters: motion to the right is positive and motion to the left is negative. This law follows directly from Newton's third law of motion: for every force one object exerts on another, there is an equal and opposite reaction force, so any momentum lost by one object is gained by the other.
Types of collisions and how to identify them
There are three main categories. In a perfectly elastic collision, both momentum and kinetic energy are conserved; hard objects like billiard balls and ideal springs approximate this behavior. In a perfectly inelastic collision (also called a completely inelastic collision), the two objects stick together after impact and move as a single body; maximum kinetic energy is lost to deformation, heat, or sound. A partially inelastic collision falls in between: momentum is conserved but some kinetic energy is lost, characterised by a coefficient of restitution between 0 and 1. An explosion is the reverse process: a stationary or slowly moving body releases stored energy and separates into two pieces; momentum is still conserved (summing to zero in the rest frame) but kinetic energy increases.
The coefficient of restitution
The coefficient of restitution (e) measures how bouncy a collision is. It is defined as the ratio of the relative speed of separation to the relative speed of approach: e = |v2 - v1| / |u1 - u2|. A value of 1 means perfectly elastic (no energy lost), a value of 0 means perfectly inelastic (objects do not separate), and any value in between describes a real-world partially inelastic collision. For example, a typical rubber ball dropped on concrete has e around 0.85 to 0.9, while a lump of clay has e near 0.
Two-dimensional collisions
This calculator handles one-dimensional (head-on) collisions where both objects move along the same line. For two-dimensional collisions, such as a glancing blow between billiard balls, the same principles apply but you must conserve momentum separately in the x-direction and the y-direction. Decompose each velocity into horizontal and vertical components, apply the conservation equation in each direction independently, then recombine the component results to find the final speed and angle of each object.
Collision types and their properties
| Collision type | Momentum conserved? | KE conserved? | e value | Example |
|---|---|---|---|---|
| Perfectly elastic | Yes | Yes | 1 | Billiard balls, ideal springs |
| Partially inelastic | Yes | No (partially lost) | 0 < e < 1 | Car crash, rubber ball |
| Perfectly inelastic | Yes | No (maximum loss) | 0 | Clay hitting wall, coupling railcars |
| Explosion / recoil | Yes | No (KE gained) | N/A | Cannon firing, air rocket |
Summary of the three main collision categories and what is conserved in each. e = coefficient of restitution.
Frequently asked questions
Why is momentum conserved but kinetic energy is not in most collisions?
Momentum is always conserved in a closed system because of Newton's third law: every action has an equal and opposite reaction, so any momentum one object gains the other loses. Kinetic energy, on the other hand, is only one form of energy. During most real collisions, some kinetic energy is converted to heat, sound, or permanent deformation of the objects. Only in an ideal elastic collision, where the objects behave like perfect springs, does kinetic energy return fully to its original form.
What is a perfectly inelastic collision?
A perfectly inelastic collision is one where the two objects stick together after impact and move as a single combined body. This produces the maximum possible loss of kinetic energy while still conserving momentum. The final shared velocity is calculated from V = (m1*u1 + m2*u2) / (m1 + m2). Common examples include a lump of clay striking a wall and sticking, a football tackle where two players grasp each other, or railcar coupling.
What happens when two equal masses collide elastically, one at rest?
When two objects of identical mass collide perfectly elastically and one is initially at rest, the moving object comes to a complete stop and the stationary object moves off at exactly the original velocity. This is because the elastic collision formulas simplify to v1 = 0 and v2 = u1 when m1 = m2 and u2 = 0. This is visibly demonstrated in Newton's cradle.
How do I use negative velocities in the calculator?
The sign convention is that motion to the right (or in the positive direction) is positive, and motion to the left (or negative direction) is negative. If object 2 is moving toward object 1 from the right, enter its initial velocity as a negative number. The calculator will correctly account for the direction and give you signed final velocities, so a negative result means the object moves to the left after the collision.
Can momentum be conserved if external forces are present?
No. Conservation of momentum applies only to isolated systems with no net external force. If a friction force, gravity, or an externally applied force acts on the system during the collision, total momentum changes. However, for most collision problems the collision duration is so short (milliseconds) that the impulse from friction or gravity is negligible compared to the collision force, so the approximation of momentum conservation is still accurate.
What is the difference between momentum and kinetic energy?
Momentum (p = mv) is a vector: it has both magnitude and direction, and it is always conserved in an isolated system. Kinetic energy (KE = 0.5*mv^2) is a scalar: it is always positive and represents the capacity of a moving object to do work. Because KE depends on the square of velocity, two objects with the same momentum but different masses have different kinetic energies. In elastic collisions both are conserved; in inelastic collisions only momentum is.