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Conservation of momentum in collisions

Consider a system of two particles A and B. Let the momentum of the particles before collision is $\displaystyle {{\overrightarrow{p}}_{A}}$ and $\displaystyle {{\overrightarrow{p}}_{B}}$ respectively and after collision let the momentum are $\displaystyle {{\overrightarrow{p'}}_{A}}$ and $\displaystyle {{\overrightarrow{p'}}_{B}}$ respectively. Let the particles are in contact for a small time Δt during collision.

Using Newton’s second law, force on A due to B :

$\displaystyle {{\overrightarrow{F}}_{AB}}=\frac{{{\overrightarrow{p'}}_{A}}-{{\overrightarrow{p}}_{A}}}{\Delta t}$ …..(1)

Similarly, force on B due to A,

$\displaystyle {{\overrightarrow{F}}_{BA}}=\frac{{{\overrightarrow{p'}}_{B}}-{{\overrightarrow{p}}_{B}}}{\Delta t}$ …..(2)

From Newton’s third law,

$\displaystyle {{\overrightarrow{F}}_{AB}}=-{{\overrightarrow{F}}_{BA}}$

∴ From equations (1) and (2),

$\displaystyle \frac{{{\overrightarrow{p'}}_{A}}-{{\overrightarrow{p}}_{A}}}{\Delta t}=-\left( \frac{{{\overrightarrow{p'}}_{B}}-{{\overrightarrow{p}}_{B}}}{\Delta t} \right)$

$\displaystyle \Rightarrow {{\overrightarrow{p}}_{A}}+{{\overrightarrow{p}}_{B}}={{\overrightarrow{p'}}_{A}}+{{\overrightarrow{p'}}_{B}}$

⇒ Total linear momentum of the system before collision = Total linear momentum of the system after collision

Conservation of momentum in collisions

Conservation of momentum in collisions

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