Momentum, Impulse & Collisions Explained | Chapter 8 of University Physics

Chapter 8 of University Physics delves into linear momentum, impulse, and the conservation of momentum—which holds even in high-speed or variable-mass scenarios. You’ll learn how to analyze collisions, track a system’s center of mass, and apply momentum concepts to rocket propulsion.

Momentum & Impulse

Linear momentum (p) is defined as p = m v and is conserved in any isolated system. As a vector, its direction matches that of velocity. Impulse (J) quantifies how force applied over time changes momentum:

• J = F Δt for constant force
• J = ∫ F dt for variable forces
• Impulse–momentum theorem: J = Δp = p₂ – p₁

Conservation of Momentum & Collisions

When net external force is zero, total momentum before equals total momentum after. This principle applies across dimensions—each component (x, y, z) conserves separately. Collisions fall into three categories:

• Elastic: both momentum and kinetic energy conserved
• Inelastic: momentum conserved, kinetic energy lost to deformation or heat
• Completely inelastic: colliding bodies stick together

In one-dimensional elastic collisions, the relative speed reverses sign but retains magnitude. Use conservation equations plus energy conditions to solve these problems.

Center of Mass Motion

The center of mass (r_cm) tracks the weighted average position of a system’s mass:

r_cm = (Σ m_i r_i) / Σ m_i

The total momentum of the system is P = M v_cm. If no net external force acts, v_cm remains constant, simplifying analyses of complex multi-particle interactions.

Rocket Propulsion & Variable Mass Systems

Rocket thrust arises from ejecting mass at high speed. Applying momentum conservation to a changing-mass system yields:

F = –v_ex (dm/dt) where v_ex is exhaust velocity.

The rocket equation relates velocity change to mass ratio:

v – v₀ = v_ex ln(m₀ / m)

This shows how final velocity depends logarithmically on initial and final mass.

Conclusion

Mastering momentum, impulse, and collision analysis equips you to solve a wide range of mechanics problems—from billiard-ball impacts to rocket launches. Tracking the center of mass and applying conservation laws simplifies even the most complex systems.

For detailed derivations and worked examples, watch the full video summary here. Be sure to explore the next chapters to deepen your physics mastery.

If you found this breakdown helpful, be sure to subscribe to Last Minute Lecture for more chapter-by-chapter textbook summaries and academic study guides.