Conservation of linear momentum
PHYS 201 · Momentum and Collisions
Total linear momentum is conserved when the net external impulse on a system is zero. This lesson explains system choice, internal forces, explosions, recoil, and vector conservation.
Key equations
\vec{P}=\sum_i \vec{p}_i=\sum_i m_i\vec{v}_i\vec{F}_{ext,net}=\frac{d\vec{P}}{dt}\frac{d\vec{P}}{dt}=0\vec{P}_i=\vec{P}_f\vec{J}_{ext}=\Delta\vec{P}m_1v_{1i}+m_2v_{2i}=m_1v_{1f}+m_2v_{2f}m_1v_{1i}+m_2v_{2i}=(m_1+m_2)v_fm_1\vec{v}_1+m_2\vec{v}_2=0\sum p_{x,i}=\sum p_{x,f}\sum p_{y,i}=\sum p_{y,f}Learning objectives
- State the condition for conservation of linear momentum.
- Distinguish internal and external forces for a chosen system.
- Apply momentum conservation in one and two dimensions.
- Analyze recoil and explosion problems using vector momentum.
Momentum of a system
For a system of particles, total momentum is the vector sum of individual momenta:
ec{P}=sum_i ec{p}_i=sum_i m_i ec{v}_i
Newton's second law for a system can be written
ec{F}_{ext,net}= rac{d ec{P}}{dt}
Only external forces can change total momentum of the system. Internal forces between parts of the system cancel in pairs by Newton's third law.
Conservation condition
If the net external force is zero, then
rac{d ec{P}}{dt}=0
so total momentum is constant:
ec{P}_i= ec{P}_f
More generally, momentum is conserved over an interval if the net external impulse is zero or negligible:
ec{J}_{ext}=Delta ec{P}
If ec{J}_{ext}=0, then Delta ec{P}=0.
Why internal forces cancel
Consider two objects colliding. Object 1 exerts a force on object 2, and object 2 exerts an equal and opposite force on object 1:
ec{F}*{1 on 2}=- ec{F}*{2 on 1}
These forces act for the same time, so their impulses are equal and opposite. They change individual momenta but not total momentum.
This is why momentum conservation is so powerful in collisions, even when the collision forces are large and complicated.
One-dimensional momentum conservation
For two objects moving along one line,
Signs indicate direction. Choosing positive consistently is essential.
If two carts stick together after collision, then
This is a perfectly inelastic collision.
Recoil
Recoil is a momentum conservation effect. Suppose a system is initially at rest, so total momentum is zero. If it separates into two parts, their momenta must add to zero:
m_1 ec{v}_1+m_2 ec{v}_2=0
Thus the parts move in opposite directions. A person jumping from a boat, a rocket expelling exhaust, and a cannon firing a projectile all involve recoil.
Explosions
In an explosion, internal energy becomes kinetic energy, but total momentum remains conserved if external impulse is negligible. A firework initially at rest breaks into fragments whose vector momenta sum to zero.
Kinetic energy usually increases in explosions because stored chemical or internal energy is converted into motion.
Momentum in two dimensions
Momentum conservation is vector conservation. In two dimensions, write separate component equations:
This is essential in glancing collisions, explosions into multiple fragments, and scattering events.
External forces during short collisions
During a short collision, external forces such as gravity may exist but contribute little impulse compared with the large internal collision forces. Momentum may then be approximately conserved during the collision, even if not conserved over longer times.
The big idea
Total linear momentum of a system is conserved when net external impulse is zero or negligible. Internal forces can redistribute momentum among parts of the system but cannot change the total. Conservation of momentum is a vector principle that explains collisions, recoil, explosions, propulsion, and many interactions where forces are difficult to track directly.
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