Conservation of momentum

In all isolated systems, total momentum is conserved:

${\overrightarrow{p}}_{\text{tot}}={\overrightarrow{p}}_{\text{tot}}^{\prime }\text{.}$

Example

Car A (2.5 kg) begins at 1.5 m/s [fwd], while Car B (1.5 kg) beings at rest. Car A collides into Car B, and they stick together. What is the velocity of AB after the collision?

We begin with the conservation of momentum,

${\overrightarrow{p}}_{\text{tot}}={\overrightarrow{p}}_{\text{tot}}^{\prime }\text{,}$

into which we substitute the individual momenta:

${\overrightarrow{p}}_{\text{A}}+{\overrightarrow{p}}_{\text{B}}={\overrightarrow{p}}_{\text{AB}}^{\prime }\text{.}$

Now we can substitute the masses and velocities with

$m_{\text{A}}{\overrightarrow{v}}_{\text{A}}+m_{\text{B}}{\overrightarrow{v}}_{\text{B}}=m_{\text{AB}}{\overrightarrow{v}}_{\text{AB}}^{\prime }\text{,}$

and solve for the final velocity:

${\overrightarrow{v}}_{\text{AB}}^{\prime }=\frac{m_{\text{A}}{\overrightarrow{v}}_{\text{A}}+m_{\text{B}}{\overrightarrow{v}}_{\text{B}}}{m_{\text{AB}}}\text{.}$

Substituting the known values where [fwd] is positive, we have

${\overrightarrow{v}}_{\text{AB}}^{\prime }=\frac{(2.5\text{kg})(1.5\text{m/s})-0}{2.5\text{kg}+1.5\text{kg}}=0.9375\text{m/s}\text{,}$

therefore the final velocity is 0.94 m/s [fwd].