Linear momentum is the product of mass and velocity, representing an object's quantity of motion. It's closely tied to impulse, which is the force applied over time that changes momentum. These concepts are crucial for understanding how objects move and interact.

The impulse-momentum theorem links these ideas, showing how forces change an object's motion. Newton's second law, rewritten in terms of momentum, further illuminates this relationship. These principles are key to analyzing collisions, explosions, and other dynamic scenarios.

Linear Momentum and Its Relationship to Force and Impulse

Concept of linear momentum

• Linear momentum ($p$) is the product of an object's mass ($m$) and its velocity ($v$) calculated using the formula $p = mv$
• Represents the quantity of motion an object possesses, considering both its mass and velocity
• Vector quantity with both magnitude and direction, meaning it has a specific value and points in a specific direction (north, east)
• Impulse ($J$) is the product of the net force ($F_{net}$) acting on an object and the time interval ($\Delta t$) over which the force acts, calculated using the formula $J = F_{net} \Delta t$
• Also a vector quantity, having both magnitude and direction (45 N·s, 60° from horizontal)
• Impulse is equal to the change in momentum ($\Delta p$) of an object, represented by the equation $J = \Delta p = p_f - p_i$, where $p_f$ is final momentum and $p_i$ is initial momentum
• Impulse can be thought of as the "kick" or "punch" that changes an object's momentum over a period of time (baseball bat hitting a ball)

Impulse-momentum theorem applications

• The impulse-momentum theorem states that the impulse applied to an object equals the change in its momentum, expressed as $J = \Delta p = p_f - p_i = m(v_f - v_i)$
• To solve problems using the impulse-momentum theorem, follow these steps:
  1. Identify the initial and final velocities ($v_i$ and $v_f$) or momenta ($p_i$ and $p_f$) of the object (car traveling at 20 m/s, coming to a stop)
  2. Determine the mass ($m$) of the object (1,500 kg car)
  3. Calculate the change in momentum ($\Delta p$) using the given information (mass and change in velocity)
  4. If the force ($F_{net}$) and time interval ($\Delta t$) are known, calculate the impulse ($J$) and equate it to the change in momentum (force of brakes over 5 seconds)
• Applying the theorem helps analyze situations involving collisions, explosions, or any scenario where forces act on objects over time (rocket launch, two billiard balls colliding)
• The coefficient of restitution can be used to characterize the elasticity of collisions in impulse-momentum problems

Newton's law and momentum change

• Newton's second law of motion states that the net force acting on an object equals the product of its mass and acceleration, expressed as $F_{net} = ma$
• The law can be rewritten in terms of momentum change: $F_{net} = \frac{\Delta p}{\Delta t}$, meaning the net force acting on an object is equal to the rate of change of its momentum
• Implications of Newton's second law in terms of momentum include:
  • If there is no net force acting on an object, its momentum remains constant, known as the conservation of momentum (spacecraft in frictionless space)
  • A net force acting on an object will cause a change in its momentum over time (thrust from a jet engine changes aircraft momentum)
  • The greater the net force, the greater the rate of change of momentum (higher force from a tennis racket causes a faster change in ball momentum)
• Newton's second law connects force, mass, and acceleration to explain changes in motion and momentum (understanding car crashes, designing safer sports equipment)
• The work-energy theorem relates the work done on an object to its change in kinetic energy, which is closely linked to momentum

Extended Concepts in Momentum

• Angular momentum is the rotational analog of linear momentum, describing the quantity of rotational motion of an object
• Momentum flux represents the flow of momentum through a surface, important in fluid dynamics and electromagnetic theory
• These concepts extend our understanding of momentum beyond linear motion to rotational systems and continuous media