Conservation of Linear Momentum - AP Physics C: Mechanics Study Guide

Getting Started

Consider a complex interaction, such as the collision of two galaxies or the explosion of a firework into many fragments. To analyze the motion of the entire collection of objects, tracking the forces on each individual piece would be impossibly complex. The principle of conservation of linear momentum provides a powerful tool to bypass these details by focusing on the system as a whole, asking a core question: Under what conditions does the total "quantity of motion" of a system remain unchanged, regardless of the intricate interactions occurring within it?

What You Should Be Able to Do

After studying this section, you will be able to:

• Define a system and distinguish between internal and external forces to determine if its total linear momentum is conserved.

• Derive the conservation of linear momentum from Newton's Second and Third Laws in differential form.

• Calculate the velocity of a system's center of mass and prove that it is constant for an isolated system.

• Apply the principle of conservation of momentum, ${\vec{P}}_{initial}={\vec{P}}_{final}$, to solve for unknown velocities in multi-dimensional collisions and explosions.

• Relate the net external force on a system to the change in its total momentum using the integral relationship $\Delta {\vec{P}}_{sys}=\int {\vec{F}}_{net,ext}dt$.

Key Concepts & Mechanisms

This topic is best understood through the lens of Dynamics as Cause, where we see how forces, or the lack thereof, cause changes in a system's momentum.

System & Preconditions

The foundational step in any momentum problem is to define the system: the collection of objects under consideration. The boundary you draw around your system is critical, as it determines which forces are internal and which are external.

• Internal Forces: Forces that objects within the system exert on each other (e.g., the collision force between two billiard balls, gravitational attraction between two stars in a binary system).

• External Forces: Forces exerted on objects in the system by agents outside the system (e.g., gravity from the Earth acting on the billiard balls, air resistance).

The crucial precondition for the conservation of linear momentum is that the system must be isolated. An isolated system is one for which the vector sum of all external forces is zero.

$$\sum {\vec{F}}_{ext}=0$$

In practice, we often treat objects as point particles or rigid bodies, ignoring internal vibrations or rotations unless they are relevant to the problem.

Key Steps / Relations

The principle of conservation of momentum is not a new, independent law of physics; rather, it is a direct and powerful consequence of Newton's Laws of Motion.

1. Start with Newton's Second Law for a single particle, written in its most general form. The net force on a particle equals the time rate of change of its linear momentum, $\vec{p}=m\vec{v}$.

   $${\vec{F}}_{net}=\frac{d\vec{p}}{dt}$$

2. Extend to a System of N Particles. The total rate of change of momentum for the system is the sum of the rates of change for each particle.

   $$\frac{d{\vec{P}}_{sys}}{dt}=\frac{d}{dt}\left(\sum _{i=1}^N{\vec{p}}_i\right)=\sum _{i=1}^N\frac{d{\vec{p}}_i}{dt}=\sum _{i=1}^N{\vec{F}}_{i,net}$$

   Here, ${\vec{P}}_{sys}$ is the total momentum of the system.

3. Separate Forces. The net force on each particle, ${\vec{F}}_{i,net}$, is the sum of external forces and internal forces.

   $$\sum _{i=1}^N{\vec{F}}_{i,net}=\sum _{i=1}^N{\vec{F}}_{i,ext}+\sum _{i=1}^N\sum _{j\ne i}{\vec{F}}_{ji}$$

   where ${\vec{F}}_{ji}$ is the internal force on particle $i$ from particle $j$.

4. Apply Newton's Third Law. For every internal force ${\vec{F}}_{ji}$, there exists an equal and opposite force ${\vec{F}}_{ij}$ that particle $i$ exerts on particle $j$. When we sum all internal forces, these action-reaction pairs cancel out perfectly.

   $$\sum _{i=1}^N\sum _{j\ne i}{\vec{F}}_{ji}=0$$

5. Arrive at the Governing Equation. With the internal forces eliminated, the governing equation for the system's dynamics simplifies to a profound statement: the rate of change of a system's total momentum is equal to the net external force acting on the system.

   $${\vec{F}}_{net,ext}=\frac{d{\vec{P}}_{sys}}{dt}$$

Outputs & Effects

The governing equation leads directly to the core principle and its consequences:

• Conservation of Momentum: If the system is isolated, ${\vec{F}}_{net,ext}=0$. The governing equation becomes $\frac{d{\vec{P}}_{sys}}{dt}=0$. The only way the derivative of a quantity can be zero is if that quantity is constant. Therefore, for an isolated system, the total linear momentum is conserved.

  $${\vec{P}}_{sys}=\text{constant}\text{or}\sum {\vec{p}}_{initial}=\sum {\vec{p}}_{final}$$

  This allows us to analyze the state of a system before and after a complex internal interaction (like a collision or explosion) without knowing the details of the internal forces.

• Center of Mass Motion: The velocity of the center of mass, ${\vec{v}}_{cm}$, is defined as the total momentum of the system divided by its total mass, $M_{total}$.

  $${\vec{v}}_{cm}=\frac{\sum m_i{\vec{v}}_i}{\sum m_i}=\frac{\sum {\vec{p}}_i}{M_{total}}=\frac{{\vec{P}}_{sys}}{M_{total}}$$

  Taking the time derivative, we find the acceleration of the center of mass: ${\vec{a}}_{cm}=\frac{d{\vec{v}}_{cm}}{dt}=\frac{1}{M_{total}}\frac{d{\vec{P}}_{sys}}{dt}$. Substituting our governing equation gives:

  $${\vec{F}}_{net,ext}=M_{total}{\vec{a}}_{cm}$$

  This reveals that the center of mass of a system moves as if it were a single point particle of mass $M_{total}$ acted upon by the net external force. If the system is isolated (${\vec{F}}_{net,ext}=0$), then ${\vec{a}}_{cm}=0$ and the velocity of the center of mass is constant.

Regulation & Limits

• Validity Domain: The law of conservation of momentum is universally valid, but its application requires a system where the net external force is genuinely zero.

• Component-wise Conservation: In many cases, a net external force may act in one direction but not another. For example, a block sliding on a frictionless horizontal surface may be subject to gravity and a normal force (vertical external forces), but no horizontal external forces. In this case, the vertical component of momentum is not conserved, but the horizontal component is.

• The Impulse Approximation: For interactions that are very strong and very brief, such as a bat hitting a ball, the internal forces of the collision are orders of magnitude larger than external forces like gravity or air resistance. During the brief interval of the collision, we can approximate the system as isolated (${\vec{F}}_{net,ext}\approx 0$) to apply momentum conservation.

Key Models & Diagrams

The decision process for applying momentum conservation can be mapped as a flowchart.

Key Components & Evidence

• Linear Momentum ($\vec{p}$): A vector quantity defined as $\vec{p}=m\vec{v}$, representing an object's inertia in motion. Its SI unit is the kg⋅m/s.

• System: A collection of objects defined by an analytical boundary. The choice of system is the most critical step in applying conservation laws.

• Internal Force: A force exerted between two objects within the chosen system. By Newton's Third Law, these forces always sum to zero across the entire system.

• External Force: A force acting on a part of the system from an agent outside the system's boundary.

• Net External Force (${\vec{F}}_{net,ext}$): The vector sum of all external forces. This is the sole cause of a change in a system's total linear momentum.

• Newton's Second Law (System Form): The fundamental differential relation ${\vec{F}}_{net,ext}=\frac{d{\vec{P}}_{sys}}{dt}$, which governs the evolution of the system's momentum.

• Conservation of Linear Momentum: A direct consequence of Newton's laws, stating that if ${\vec{F}}_{net,ext}=0$, then the total momentum vector ${\vec{P}}_{sys}$ of the system is constant in time.

• Center of Mass Velocity (${\vec{v}}_{cm}$): The velocity of a fictitious point that moves as if all the system's mass were concentrated there. It is defined as ${\vec{v}}_{cm}=\frac{{\vec{P}}_{sys}}{\sum m_i}$. For an isolated system, ${\vec{v}}_{cm}$ is constant.

Skill Snapshots

Causation

• Driver: A non-zero net external force on a system. → Change: The system's total linear momentum changes at a rate equal to that force ($\frac{d{\vec{P}}_{sys}}{dt}={\vec{F}}_{net,ext}$).

• Driver: A zero net external force on a system. → Change: The system's total linear momentum remains constant ($\frac{d{\vec{P}}_{sys}}{dt}=0$).

• Driver: An internal explosion within an isolated system. → Change: The momenta of the individual fragments change dramatically, but in such a way that their vector sum remains constant.

Comparison

• An isolated system (${\vec{F}}_{net,ext}=0$) maintains a constant total momentum, whereas a non-isolated system (${\vec{F}}_{net,ext}\ne 0$) experiences a change in total momentum.

• The momentum of a single particle in a collision changes due to the large (external) collision force, while the momentum of the system of both particles remains constant because the collision force is internal.

• Conservation of Momentum is a vector principle applicable to all interactions in an isolated system, whereas Conservation of Kinetic Energy is a scalar principle that applies only to perfectly elastic collisions.

Change and Continuity Over Time

• Baseline: A rocket of mass $M$ and its fuel travel together through space with an initial velocity ${\vec{v}}_i$, giving the system an initial momentum ${\vec{P}}_{sys,i}=M{\vec{v}}_i$.

• Change: The rocket expels hot gas (a part of the system) backward with high velocity. This is an internal process driven by internal forces.

• Change: To conserve the system's total momentum, the rocket (the other part of the system) must gain an equal and opposite amount of momentum, causing it to accelerate forward.

• Continuity: If there are no external forces (like drag or gravity), the center of mass of the rocket-plus-exhaust-gas system continues to move forward with the original, constant velocity ${\vec{v}}_i$.

Common Misconceptions & Clarifications

1. Misconception: "Momentum is always conserved in a collision."

   • Clarification: The total momentum of the system of all colliding objects is conserved, provided there are no significant net external forces (like friction). The momentum of any single object involved is certainly not conserved; it changes drastically. The choice of system is paramount.

2. Misconception: "If the total momentum of a system is conserved, its kinetic energy must also be conserved."

   • Clarification: This is incorrect. In an inelastic collision (e.g., two clay balls sticking together), momentum is conserved, but kinetic energy is converted into thermal energy and sound. Conservation of momentum is more robust than conservation of mechanical energy.

3. Misconception: "If momentum is conserved, the center of mass must be stationary."

   • Clarification: Conserved means constant, not necessarily zero. If an isolated system is already in motion, its total momentum is non-zero and constant. Consequently, its center of mass moves with a constant, non-zero velocity.

4. Misconception: "Internal forces don't affect the motion of objects."

   • Clarification: Internal forces are responsible for all the interesting changes to the individual objects within a system—they cause fragments of an explosion to fly apart and cars to crumple in a collision. They simply cannot change the motion of the system's center of mass.

One-Paragraph Summary

The principle of conservation of linear momentum is a powerful consequence of Newton's Second and Third Laws, providing a method to analyze complex interactions. By defining a system of objects, we can distinguish between internal forces, which cancel out in pairs, and external forces. The governing law for a system is that the net external force equals the time rate of change of the system's total momentum, ${\vec{F}}_{net,ext}=d{\vec{P}}_{sys}/dt$. In the crucial case of an isolated system where the net external force is zero, the total momentum is conserved—it remains constant over time. This implies that the system's center of mass moves with a constant velocity, regardless of the collisions, explosions, or other intricate interactions occurring within the system. This principle allows us to predict the outcomes of such events without needing to analyze the complex, transient internal forces involved.