AcademyMomentum Systems
Academy
Collisions as Momentum Problems
Level 1 - Physics topic page in Momentum Systems.
Principle
A collision is a short interaction where momentum is modeled before kinetic energy.
Notation
\(m_1,m_2\)
colliding masses
\(\mathrm{kg}\)
\(u_1,u_2\)
initial velocities
\(\mathrm{m\,s^{-1}}\)
\(v_1,v_2\)
final velocities
\(\mathrm{m\,s^{-1}}\)
\(\vec{J}_{12}\)
impulse on 1 by 2
\(\mathrm{N\,s}\)
\(K\)
kinetic energy
\(\mathrm{J}\)
Method
During a short collision, internal impulses can be large while external impulse is often negligible.
Choose interval
\[\Delta t_{\mathrm{collision}}\ \text{is short}\]
Neglect external impulse
\[\Delta\vec{P}_{\mathrm{system}}\approx\vec{0}\]
Write momentum
\[m_1u_1+m_2u_2=m_1v_1+m_2v_2\]
Check energy
\[\Delta K=K_f-K_i\]
Use this as a classification check unless the collision is specified elastic.
The diagram compares the total momentum immediately before and after the short interaction.
The before-after momentum vectors match for an isolated collision, but kinetic energy may not.
Rules
These are the compact one-dimensional collision relations.
Momentum equation
\[m_1u_1+m_2u_2=m_1v_1+m_2v_2\]
Sticking collision
\[m_1u_1+m_2u_2=(m_1+m_2)v_f\]
Impulse pair
\[\vec{J}_{12}=-\vec{J}_{21}\]
Energy change
\[\Delta K=K_f-K_i\]
Examples
Question
A
\[0.80\,\mathrm{kg}\]
cart moving at \[5.0\,\mathrm{m\,s^{-1}}\]
sticks to a \[1.2\,\mathrm{kg}\]
cart at rest. Find final speed.Answer
\[0.80(5.0)=(0.80+1.2)v_f\]
\[v_f=2.0\,\mathrm{m\,s^{-1}}\]
Checks
- Momentum conservation is the default collision equation for an isolated pair.
- Kinetic energy is conserved only for elastic collisions.
- A stuck-together final state is not elastic.
- Short collision time does not remove large internal impulses.