Linear momentum is a crucial concept in physics, describing an object's motion based on its mass and velocity. It's calculated by multiplying these two factors, resulting in a vector quantity with both magnitude and direction. Understanding momentum is key to grasping how objects interact and move.

Momentum plays a central role in Newton's laws of motion and the principle of conservation. In collisions and interactions, the total momentum of a closed system remains constant. This fundamental concept helps explain everything from everyday collisions to complex particle interactions in physics.

Linear Momentum

Definition of linear momentum

• Linear momentum ($p$) calculated by multiplying an object's mass ($m$) and velocity ($v$)
  • Formula: $p = mv$
  • SI unit: kilogram-meter per second (kg⋅m/s)
• Larger mass or velocity results in greater momentum (bowling ball, bullet)
• Momentum is a vector quantity with both magnitude and direction
  • Momentum direction same as velocity direction (car moving north)
• Center of mass is the point where the entire mass of an object can be considered concentrated for momentum calculations

Momentum in Newton's second law

• Newton's second law: net force ($F_{net}$) equals rate of change of momentum ($dp/dt$)
  • Formula: $F_{net} = dp/dt$
  • For constant mass: $F_{net} = m(dv/dt) = ma$, $a$ is acceleration
• Net force acting on an object changes its momentum (pushing a shopping cart)
  • Change in momentum directly proportional to net force and time interval ($\Delta t$)
    • Formula: $\Delta p = F_{net} \Delta t$
• Impulse ($J$) is product of net force and time interval
  • Formula: $J = F_{net} \Delta t = \Delta p$
  • SI unit: newton-second (N⋅s)
  • Impulse equals change in momentum (hitting a tennis ball)
• Work-energy theorem relates the work done by net force to change in kinetic energy, complementing momentum analysis

Conservation of Momentum

Momentum analysis in collisions

• Colliding or interacting objects experience momentum changes due to mutual forces
• Elastic collisions: conserved kinetic energy, total momentum remains constant
  • Examples: billiard balls, certain atomic and subatomic particle interactions
• Inelastic collisions: kinetic energy not conserved, total momentum remains constant
  • Examples: colliding vehicles, clay ball hitting a wall
• Perfectly inelastic collisions: objects stick together after collision with common velocity
  • Maximum kinetic energy loss in these collisions (two lumps of clay colliding)
• Coefficient of restitution measures the elasticity of a collision, ranging from 0 (perfectly inelastic) to 1 (perfectly elastic)

Conservation of momentum in systems

• Law of conservation of momentum: total momentum of closed system remains constant
  • Closed system has no external forces acting on objects within (space probe)
• Without external forces, total momentum before interaction equals total momentum after
  • Formula: $m_1v_1 + m_2v_2 = m_1v'_1 + m_2v'_2$, $v$ and $v'$ are initial and final velocities
• Conservation of momentum solves problems involving collisions and explosions (rocket launch)
  • Equating total momentum before and after interaction calculates unknown velocities or masses
• Law of conservation of momentum is fundamental physics principle for all isolated systems (particles to planets)

Force and Momentum Interactions

• Newton's third law states that for every action, there is an equal and opposite reaction
• Momentum transfer occurs when objects interact, exchanging momentum between them
• The total momentum of the system remains constant due to these equal and opposite forces