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In an isolated system (no net external force), the total linear momentum is constant: Σp_i = Σp_f. This follows directly from Newton's third law: internal forces between objects in the system are equal and opposite, so they cancel in the sum. Conservation of momentum is valid even when energy is not conserved (inelastic collisions), making it an indispensable tool.
Apply in both 1D (one equation) and 2D (two component equations) scenarios. Verify the direction of each momentum vector carefully and use consistent sign conventions.
Conservation of momentum is not a separate law tacked onto Newton's mechanics — it follows directly from Newton's third law, which you already know. During any collision or interaction between two objects A and B, the force A exerts on B is equal and opposite to the force B exerts on A. These forces act for exactly the same duration (they're simultaneous), so the impulses they deliver are equal and opposite. But impulse equals change in momentum (from your work on momentum and impulse), so the change in momentum of A exactly cancels the change in momentum of B. The total momentum of the system — Σp = p_A + p_B — does not change.
The crucial condition is *isolation*: the system must experience no net external force. Internal forces (between objects within the system) always cancel in pairs by Newton's third law. External forces don't cancel this way. Friction with the ground, air resistance, a rope attached to a wall — any of these can inject momentum into the system from outside and break conservation. When you set up a momentum problem, your first step should always be identifying the system and checking whether external horizontal (or vertical, depending on the setup) forces are negligible. Frictionless ice, negligible air resistance, or very short collision times (where external impulse during the collision is small) all justify treating the system as isolated.
What makes conservation of momentum especially powerful is that it works even when kinetic energy is not conserved. In a perfectly inelastic collision where two objects stick together, kinetic energy is lost to heat and deformation — but momentum is still conserved. Many students conflate these two conservation laws and conclude that momentum can't be conserved in "messy" collisions. That's wrong: they are independent. Momentum conservation holds whenever there's no net external impulse; energy conservation (in its kinetic-energy form) holds only for elastic collisions. In most real collisions, use momentum conservation confidently and treat energy as a separate question.
In two dimensions, momentum is a vector equation, which means you get two independent equations — one for each component. If a ball moving east hits a stationary ball and they scatter at angles, the x-component of total momentum is conserved and the y-component is conserved separately. Write the vector equation first (Σp_i = Σp_f), then project onto x and y axes. The most common error in 2D is treating momentum as a scalar — adding magnitudes instead of components — which gives wrong answers whenever the objects don't move along the same line.
Finally, note the connection to the center of mass. For an isolated system, the center of mass moves at constant velocity (including staying stationary if it starts at rest). This is another way of saying total momentum is conserved: Σp = MV_cm, and if Σp doesn't change, neither does V_cm. This framing is useful in explosion problems where a stationary object breaks apart — the center of mass stays put even as the pieces fly outward.