Linear Momentum and Collisions
Linear momentum is the product of mass and velocity; its conservation for isolated systems makes it the key tool for analyzing collisions and the motion of systems of particles.
Definition
Linear momentum is the vector quantity p = mv whose total is conserved for a system free of external forces; impulse is the time integral of force and equals the change in momentum, and collisions are interactions analyzed through momentum conservation.
Scope
This topic covers linear momentum, impulse, the center of mass and its motion, conservation of total momentum for systems with no external force, and the classification of collisions as elastic or inelastic. It includes variable-mass problems such as rocket motion.
Core questions
- Why is total momentum conserved when no external force acts on a system?
- How does impulse relate force and the change of momentum over time?
- How do elastic and inelastic collisions differ in what is conserved?
Key concepts
- Linear momentum
- Impulse
- Center of mass and its motion
- Elastic and inelastic collisions
- Coefficient of restitution
- Variable-mass (rocket) systems
Key theories
- Conservation of linear momentum
- For a system with no net external force, the total linear momentum is constant in time, following from Newton's third law applied to internal interaction forces.
- Impulse-momentum theorem
- The impulse delivered to a body, the time integral of the net force, equals the body's change in linear momentum, which is especially useful for short, intense collision forces.
Clinical relevance
Momentum and collision analysis underlie vehicle crash safety and crumple-zone design, ballistics, propulsion and rocketry, and the interpretation of scattering experiments, anywhere short-duration interactions transfer motion between bodies.
History
The conservation of momentum in collisions was established in the 1660s by Huygens, Wallis, and Wren, who corrected Descartes's earlier scalar conception of conserved motion by recognizing momentum as a directed vector quantity. Newton incorporated these collision results into the Principia, and the principle later generalized to systems and continua.
Key figures
- Isaac Newton
- Christiaan Huygens
- John Wallis
Related topics
Seminal works
- kleppner2014
- goldstein2002
Frequently asked questions
- Is kinetic energy conserved in all collisions?
- No. Momentum is conserved in any collision with no external force, but kinetic energy is conserved only in perfectly elastic collisions; inelastic collisions convert some kinetic energy to heat or deformation.
- Why does the center of mass move at constant velocity in an isolated system?
- Because the total external force is zero, the total momentum is constant, and the center-of-mass velocity equals total momentum divided by total mass, which therefore does not change.