Conservation of Momentum for Isolated Systems — AP Physics 1
1. Core Law of Conservation of Momentum ★★☆☆☆ ⏱ 3 min
Conservation of momentum is one of three fundamental conservation laws tested in AP Physics 1 Unit 5, making up 12–18% of total exam score. It appears in both multiple-choice and free-response sections, often combined with energy conservation or kinematics for multi-step problems.
p_{total, initial} = p_{total, final} = \sum p_i = \sum p_f
Since momentum is a vector quantity, the law holds component-wise. This vector nature is the most commonly tested feature on the AP Physics 1 exam.
2. Classifying Internal vs External Forces ★★☆☆☆ ⏱ 4 min
To apply conservation of momentum, your first step in any problem is to correctly define your system and classify forces as internal or external. This is the most frequently tested skill in AP momentum problems.
From the impulse-momentum theorem, we derive the conservation law:
\Delta p_{total} = J_{net} = \sum F_{ext} \Delta t \implies \sum F_{ext} = 0 \rightarrow \Delta p_{total} = 0
Exam tip: Never assume a system is fully isolated across all directions. Always check net external force direction by direction — AP exam questions intentionally design problems where momentum is only conserved in one direction to test this skill.
3. One-Dimensional Collisions ★★★☆☆ ⏱ 3 min
Collisions are the most common context for momentum conservation on the AP exam. During a collision, interaction time between objects is very short, so even if a small external force (like friction) acts, the impulse from the external force is negligible compared to the impulse from large internal collision forces. We almost always treat the system of colliding objects as isolated during the collision.
For two objects colliding along a straight line, the conservation equation becomes:
m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}
All velocities are signed based on your chosen positive direction. Critically, this equation applies to *all* collisions (both elastic and inelastic) for isolated systems — only kinetic energy changes between collision types, momentum is always conserved.
Exam tip: The answer zero is a perfectly valid result for momentum problems! Don't automatically assume you made a mistake if your final velocity is zero — it just means the initial momenta of the two objects canceled out exactly.
4. Conservation of Momentum in Explosions ★★★☆☆ ⏱ 3 min
Explosions are the reverse of perfectly inelastic collisions: a single object splits into two or more fragments due to internal forces from released stored energy (chemical, elastic potential, etc.). Like collisions, explosions happen over a very short time, so external impulse is negligible, and momentum is conserved for the system of fragments.
A common exam scenario is an object that is momentarily at rest before exploding, so total initial momentum is zero. This gives the simplified equation:
0 = m_1 v_{1f} + m_2 v_{2f} \implies v_{2f} = -\frac{m_1}{m_2} v_{1f}
The negative sign confirms the two fragments move in opposite directions. Unlike collisions, explosions always have an increase in total kinetic energy, as stored potential energy is converted to motion of the fragments.
Exam tip: Always check if the original object is moving or at rest before the explosion. Don't default to zero initial momentum if the problem states the object is moving horizontally before exploding.
5. Center of Mass Velocity for Isolated Systems ★★★☆☆ ⏱ 3 min
A key conceptual result of momentum conservation is that the velocity of the center of mass ($v_{CM}$) of an isolated system never changes, even if objects inside the system move relative to each other. The formula for center of mass velocity is:
v_{CM} = \frac{\sum m_i v_i}{M_{total}} = \frac{p_{total}}{M_{total}}
Since $p_{total}$ is constant for an isolated system, $v_{CM}$ must also be constant. If the system is initially at rest, $v_{CM} = 0$ forever as long as the system stays isolated. This is a great check for your calculations: if your final $v_{CM}$ doesn't match the initial $v_{CM}$, you made an algebra or sign error.
Exam tip: For conceptual FRQ questions asking why $v_{CM}$ doesn't change, always structure your answer as: 1) The system is isolated, so $\sum F_{ext} = 0$; 2) Therefore total momentum is conserved; 3) $v_{CM} = p_{total}/M_{total}$, so $v_{CM}$ is constant. This is the exact reasoning AP graders look for.
6. Concept Check ★★★☆☆ ⏱ 2 min
Common Pitfalls
Why: Students forget that Earth is outside the system unless explicitly included, so all forces from Earth are external
Why: Students assume momentum is always conserved, regardless of system choice
Why: Students treat momentum as a scalar instead of a vector, and add all speeds regardless of direction
Why: Students confuse conservation during the collision event with conservation after, when external forces like friction change momentum
Why: Students confuse the two conservation laws, and assume both hold for all isolated systems