AcademyMomentum Systems
Academy
Momentum Conservation
Level 1 - Physics topic page in Momentum Systems.
Principle
Total momentum changes only by the external impulse on the chosen system.
Notation
\(\vec{P}\)
total system momentum
\(\mathrm{kg\,m\,s^{-1}}\)
\(\vec{p}_i\)
momentum of object i
\(\mathrm{kg\,m\,s^{-1}}\)
\(\vec{J}_{\mathrm{ext}}\)
external impulse
\(\mathrm{N\,s}\)
\(m_i\)
mass of object i
\(\mathrm{kg}\)
\(\vec{v}_i\)
velocity of object i
\(\mathrm{m\,s^{-1}}\)
Method
The system boundary decides which impulses are internal and which can change total momentum.
Add momenta
\[\vec{P}=\sum_i\vec{p}_i=\sum_i m_i\vec{v}_i\]
Sum impulses
\[\Delta\vec{P}=\vec{J}_{\mathrm{internal}}+\vec{J}_{\mathrm{ext}}\]
Cancel internal pairs
\[\vec{J}_{\mathrm{internal}}=\vec{0}\]
Internal third-law impulses cancel in the total system sum.
Test isolation
\[\vec{J}_{\mathrm{ext}}=\vec{0}\Rightarrow\vec{P}_f=\vec{P}_i\]
The vector diagram below shows individual momenta adding to one total momentum vector.
Conservation means this total vector is unchanged, even when the individual vectors redistribute.
Rules
These are the compact system-momentum statements.
Total momentum
\[\vec{P}=\sum_i \vec{p}_i=\sum_i m_i\vec{v}_i\]
System impulse
\[\Delta\vec{P}=\vec{J}_{\mathrm{ext}}\]
Conservation
\[\vec{P}_f=\vec{P}_i\quad(\vec{J}_{\mathrm{ext}}=\vec{0})\]
Component form
\[\sum p_{ix}=\sum p_{fx},\qquad \sum p_{iy}=\sum p_{fy}\]
Examples
Question
A
\[1.5\,\mathrm{kg}\]
cart moving right at \[4.0\,\mathrm{m\,s^{-1}}\]
sticks to a \[2.5\,\mathrm{kg}\]
cart moving left at \[0.80\,\mathrm{m\,s^{-1}}\]
Find their common velocity.Answer
Take right as positive.
\[1.5(4.0)+2.5(-0.80)=(4.0)v_f\]
\[v_f=1.0\,\mathrm{m\,s^{-1}}\]
right.Checks
- Conservation applies to the chosen system, not automatically to each object.
- Internal forces change individual momenta but not total system momentum.
- Momentum can be conserved while kinetic energy changes.
- Use components when directions are not collinear.