Conservation of Momentum in 2D

At GCSE, conservation of momentum problems were one-dimensional — objects colliding or exploding along a single line. At A-Level, you must handle collisions where objects move in two dimensions. The key insight is that momentum is conserved independently along each axis.

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Momentum as a Vector

Momentum p = mv is a vector. This means it has both magnitude and direction. When we say momentum is conserved, we mean the total vector momentum of the system is unchanged — not just its magnitude. In two dimensions, this means:

$\sum p_x \text{ (before)} = \sum p_x \text{ (after)}$∑px​ (before)=∑px​ (after) $\sum p_y \text{ (before)} = \sum p_y \text{ (after)}$∑py​ (before)=∑py​ (after)

Conservation of momentum applies separately along the x-axis and the y-axis. This gives us two independent equations, which is what makes 2D problems solvable.

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Resolving Momentum into Components

If an object of mass m moves at speed v at angle θ to the x-axis, its momentum components are:

$p_x = mv\cos\theta$px​=mvcosθ $p_y = mv\sin\theta$py​=mvsinθ

This is exactly the same as resolving any vector into perpendicular components.

Worked Example 1: Snooker Ball Collision