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When a cue ball strikes an object ball off-center on a pool table, they ricochet away at angles instead of traveling along a single straight line.
How do physicists predict the exact final speeds and trajectory angles of both balls after such a glancing collision?

Momentum as a Vector

Linear momentum is the product of an object's mass and its velocity vector: p⃗=mv⃗\vec{p} = m\vec{v}. Because velocity has direction, momentum is strictly a vector quantity.
Any vector in two dimensions splits cleanly into two perpendicular components using right-triangle trigonometry: px=mvcos⁑θp_x = m v \cos\theta and py=mvsin⁑θp_y = m v \sin\theta.
In the 1650s, Dutch physicist Christiaan Huygens demonstrated that momentum is conserved in collisions, paving the way for Newton to formalize that perpendicular spatial dimensions act completely independently.
Why can we treat the horizontal and vertical motions as two separate problems that never interfere with each other?

Independent Component Equations

According to Newton's Third Law, during a collision object 1 exerts a contact force Fβƒ—21\vec{F}_{21} on object 2, and object 2 exerts an equal and opposite force Fβƒ—12=βˆ’Fβƒ—21\vec{F}_{12} = -\vec{F}_{21}.
Because forces along the x-axis produce zero acceleration along the y-axis, the total momentum along each axis is independently conserved:
βˆ‘pix=βˆ‘pfxβ€…β€ŠβŸΉβ€…β€Šm1v1ix+m2v2ix=m1v1fx+m2v2fx\sum p_{ix} = \sum p_{fx} \implies m_1 v_{1ix} + m_2 v_{2ix} = m_1 v_{1fx} + m_2 v_{2fx}