Conservation of Momentum — Elastic & Inelastic Collisions

Conservation of momentum means the total momentum of a system stays constant if the net external impulse on that system is zero or negligible over the time interval you care about. In collision problems, that is the rule that connects the motion before impact to the motion after impact.

Momentum is a vector quantity. For an object with constant mass in ordinary introductory mechanics, its momentum is

$$\vec{p} = m\vec{v}$$

Because momentum has direction, signs or vector directions matter. In a one-dimensional problem, motion to the right is often taken as positive and motion to the left as negative.

What Conservation Of Momentum Means In Collisions

Momentum is not automatically conserved for every single object by itself. It is conserved for the total system when the net external impulse on that system is negligible.

That condition matters. During a short collision, the forces between the colliding objects can be very large, but those are internal forces if both objects are part of the same system. Internal forces can rearrange momentum between the objects without changing the total.

For a two-object system in one dimension, the momentum statement is

$$m_1 v_{1,i} + m_2 v_{2,i} = m_1 v_{1,f} + m_2 v_{2,f}$$

This equation is valid when the system is isolated well enough over the collision interval.

Elastic Vs Inelastic Collisions

The key difference is not whether momentum is conserved. For an isolated system, momentum is conserved in both.

An elastic collision also conserves kinetic energy:

$$\frac{1}{2}m_1 v_{1,i}^2 + \frac{1}{2}m_2 v_{2,i}^2 = \frac{1}{2}m_1 v_{1,f}^2 + \frac{1}{2}m_2 v_{2,f}^2$$

An inelastic collision does not conserve total kinetic energy, even though total momentum is still conserved.

A perfectly inelastic collision is the special case where the objects stick together after impact. Then they share one final velocity.

Worked Example: A Perfectly Inelastic Collision

A $2\ \mathrm{kg}$ cart moves right at $4\ \mathrm{m/s}$ and collides with a $1\ \mathrm{kg}$ cart at rest. After the collision, the carts stick together. Find their final velocity.

This is a perfectly inelastic collision, so momentum is conserved and both carts have the same final velocity $v_f$.

Before the collision,

$$p_i = (2)(4) + (1)(0) = 8\ \mathrm{kg \cdot m/s}$$

After the collision, the combined mass is $2 + 1 = 3\ \mathrm{kg}$, so

$$p_f = (3)v_f$$

Set initial and final momentum equal:

$$8 = 3v_f$$

So the final velocity is

$$v_f = \frac{8}{3}\ \mathrm{m/s}$$

That is about $2.67\ \mathrm{m/s}$ to the right.

Now compare the kinetic energy before and after the collision:

$$K_i = \frac{1}{2}(2)(4^2) = 16\ \mathrm{J}$$ $$K_f = \frac{1}{2}(3)\left(\frac{8}{3}\right)^2 = \frac{32}{3}\ \mathrm{J} \approx 10.67\ \mathrm{J}$$

The total momentum stays the same, but the kinetic energy decreases. That is exactly what you should expect in a perfectly inelastic collision.

Why Kinetic Energy Can Decrease In An Inelastic Collision

In an inelastic collision, some kinetic energy is transformed into other forms such as thermal energy, sound, or internal deformation. That does not violate momentum conservation.

The common mistake is to assume that if one quantity is conserved, then every familiar quantity must also stay the same. Conservation laws have different conditions and apply to different physical quantities.

Common Mistakes In Momentum Problems

Treating momentum like a plain number

Momentum has direction. In one dimension, choosing a sign convention is essential. In two or three dimensions, you must conserve momentum component by component.

Forgetting to define the system

If you track only one object in a collision, its momentum usually changes. The conservation law applies to the total momentum of the isolated system, not necessarily to each object separately.

Using kinetic energy conservation for every collision

That is only valid for elastic collisions. For inelastic collisions, use momentum conservation first and add only the conditions that actually apply.

Ignoring external impulse

If outside forces matter over the time interval, total momentum for your chosen system may not stay constant. The isolation condition is part of the rule, not an optional detail.

Where Conservation Of Momentum Is Used

Conservation of momentum is used in collision analysis, recoil problems, explosions, particle interactions, and many lab-cart experiments. The same principle helps you understand why a firearm recoils, why billiard balls exchange motion, and why two objects that stick together move more slowly than the faster object did before impact.

Try A Similar Momentum Problem

Keep the same masses but let the second cart move left before the collision, then recompute the total initial momentum with a negative sign. If you want to try your own version with different numbers, solve a similar collision with GPAI Solver.