Zero Net External Force: Total momentum of the system stays constant. Nonzero Net External Force: Momentum is transferred between the system and its surroundings.

Elastic Collision: Kinetic energy is conserved. Inelastic Collision: Kinetic energy is not conserved.

1. Identify all objects in the system. 2. Determine the mass and velocity of each object. 3. Calculate the sum of the individual momenta (mᵢvᵢ). 4. Calculate the sum of the individual masses (mᵢ). 5. Divide the total momentum by the total mass: $\vec{v}_{\mathrm{cm}}=\frac{\sum \vec{p}_{i}}{\sum m_{i}}=\frac{\sum\left(m_{i} \vec{v}_{i}\right)}{\sum m_{i}}$

1. Define the system. 2. Check for external forces. If net external force is zero, momentum is conserved. 3. Write the conservation of momentum equation: Total momentum before = Total momentum after. 4. Solve for the unknown variable(s).

1. Choose your system carefully. 2. Apply conservation of momentum along each axis (x and y). 3. Account for the vector nature of momentum, using positive and negative signs for directions. 4. Solve the resulting equations for the unknowns.

1. Calculate the total kinetic energy before the collision. 2. Calculate the total kinetic energy after the collision. 3. Compare the initial and final kinetic energies. 4. If kinetic energy is conserved (KE_initial = KE_final), the collision is elastic. If kinetic energy is not conserved, the collision is inelastic.

1. Identify the force acting on the object and the time interval over which it acts. 2. If the force is constant, multiply the force by the time interval: J = FΔt. 3. If the force is not constant, integrate the force with respect to time: J = ∫F dt. 4. Alternatively, calculate the change in momentum: J = Δp = mv_f - mv_i.

Elastic Collision: Kinetic energy is conserved; objects bounce off each other. | Inelastic Collision: Kinetic energy is not conserved; some energy is converted to other forms (e.g., heat, sound); objects may stick together.

Zero Net External Force: Total momentum of the system remains constant; momentum is conserved within the system. | Nonzero Net External Force: Momentum can be transferred between the system and its surroundings; momentum is not conserved within the system alone.

Momentum: A vector quantity; conserved in a closed system regardless of elasticity of collision. | Kinetic Energy: A scalar quantity; conserved only in elastic collisions.

Impulse: The change in momentum of an object. It is a force acting over a period of time. | Momentum: The mass of an object multiplied by its velocity. It is a measure of how difficult it is to stop a moving object.

1D Collisions: Momentum conservation applied along a single axis; simpler algebraic solutions. | 2D Collisions: Momentum conservation applied along two axes (x and y); requires vector decomposition and potentially more complex algebraic solutions.