⚙️AP Physics C: Mechanics Unit 4 – Systems of Particles & Linear Momentum

Key Concepts

• Systems of particles consist of multiple objects that interact with each other through forces and can be treated as a single entity
• Center of mass is the point where the weighted relative position of the distributed mass sums to zero
• Linear momentum is the product of an object's mass and velocity ($p = mv$)
• Conservation of momentum states that the total momentum of a closed system remains constant over time
• Elastic collisions conserve both momentum and kinetic energy, while inelastic collisions only conserve momentum
• Rocket propulsion relies on the conservation of momentum, expelling mass in one direction to accelerate in the opposite direction
• Problem-solving strategies for systems of particles include defining the system, identifying conserved quantities, and applying conservation laws

Defining the System

• Clearly identify the objects that make up the system and their relevant properties (mass, velocity, etc.)
• Determine whether the system is closed (no external forces) or open (external forces present)
  • Closed systems have constant total momentum
  • Open systems may have changing total momentum due to external forces
• Consider the time interval over which the system is being analyzed
• Identify any constraints or connections between objects in the system
• Establish a coordinate system and reference frame for describing the motion of the system
• Determine if the system can be treated as a point particle or if its size and shape are relevant
• Assess whether the system is isolated or interacts with its surroundings

Center of Mass

• The center of mass is the point that represents the average position of the system's mass
• For a system of particles with masses $m_i$ and positions $\vec{r}_i$, the center of mass position is given by:
  • $\vec{r}_{CM} = \frac{\sum_i m_i \vec{r}_i}{\sum_i m_i}$
• The center of mass moves as if all the system's mass were concentrated at that point and all external forces were applied there
• For a continuous object, the center of mass is calculated using an integral:
  • $\vec{r}_{CM} = \frac{\int \vec{r} dm}{M}$
• The motion of the center of mass depends only on the net external force, not on internal forces within the system
• In the absence of external forces, the center of mass moves with constant velocity
• The center of mass is not always located within the physical boundaries of the system (e.g., a hollow sphere)

Linear Momentum

• Linear momentum is a vector quantity defined as the product of an object's mass and velocity ($\vec{p} = m\vec{v}$)
• The total momentum of a system is the vector sum of the individual momenta:
  • $\vec{P} = \sum_i \vec{p}_i = \sum_i m_i \vec{v}_i$
• The net external force on a system equals the rate of change of its total momentum (Newton's Second Law):
  • $\vec{F}_{net} = \frac{d\vec{P}}{dt}$
• In the absence of external forces, the total momentum of a system is conserved
• Momentum is a conserved quantity in all three dimensions
• The impulse-momentum theorem relates the change in momentum to the impulse ($\vec{J} = \Delta \vec{p}$)
  • Impulse is the product of the average force and the time interval over which it acts

Conservation of Momentum

• In a closed system, the total momentum remains constant over time ($\vec{P}_i = \vec{P}_f$)
• Conservation of momentum applies to systems with no net external force
• For a collision between two objects, the conservation of momentum equation is:
  • $m_1 \vec{v}_{1i} + m_2 \vec{v}_{2i} = m_1 \vec{v}_{1f} + m_2 \vec{v}_{2f}$
• Conservation of momentum is independent of the type of collision (elastic or inelastic)
• Internal forces within the system do not affect the total momentum
• Conservation of momentum can be used to analyze explosions, where a single object separates into multiple fragments
• In two dimensions, conservation of momentum applies to both the x and y components independently

Collisions

• Collisions involve two or more objects interacting through forces for a short time
• Elastic collisions conserve both momentum and kinetic energy
  • Kinetic energy before and after the collision is the same ($KE_i = KE_f$)
  • Objects separate after the collision
• Inelastic collisions conserve momentum but not kinetic energy
  • Some kinetic energy is converted to other forms (heat, deformation, etc.)
  • Objects may stick together after the collision (perfectly inelastic)
• The coefficient of restitution ($e$) characterizes the elasticity of a collision
  • $e = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}$
  • $e = 1$ for perfectly elastic collisions, $0 < e < 1$ for partially inelastic collisions, and $e = 0$ for perfectly inelastic collisions
• In two-dimensional collisions, the initial and final momenta can be resolved into components
• The impulse experienced by each object in a collision is equal and opposite

Rocket Propulsion

• Rockets accelerate by expelling mass (propellant) in one direction, causing the rocket to accelerate in the opposite direction
• The rocket and the expelled propellant form a closed system, so momentum is conserved
• The rocket's acceleration depends on the rate at which mass is expelled and the velocity of the exhaust relative to the rocket
• The rocket equation describes the motion of a rocket:
  • $\Delta v = v_e \ln \frac{m_i}{m_f}$
  • $\Delta v$ is the change in velocity, $v_e$ is the exhaust velocity, $m_i$ is the initial mass, and $m_f$ is the final mass
• The thrust force on the rocket is equal to the rate of change of momentum of the expelled propellant
• Rocket propulsion is an example of an open system, as mass is leaving the system
• The efficiency of a rocket depends on the specific impulse, which is related to the exhaust velocity
• Multi-stage rockets are used to increase payload capacity and efficiency by discarding empty fuel tanks

Problem-Solving Strategies

• Identify the system and draw a clear diagram showing the initial and final states
• Determine if the system is closed or open, and identify any external forces
• Choose a convenient coordinate system and reference frame
• Write down the relevant equations, such as conservation of momentum and kinetic energy
• If the system is constrained or connected, consider the equations of constraint
• Identify the known and unknown quantities, and solve for the desired variables
• Check if the answer is reasonable and consistent with the given information
• Consider special cases, such as elastic or perfectly inelastic collisions, to simplify the problem
• Use symmetry arguments when applicable to reduce the number of unknowns
• Break complex problems into smaller sub-problems and solve them step by step