Linear Momentum · ⁨Động lượng tuyến tính⁩

Learning Objective 4.1.A: Describe the linear momentum of an object or system.

• 4.1.A.1 Linear momentum is defined by the equation $\vec{p} = m\vec{v}$.
• 4.1.A.2 Momentum is a vector quantity and has the same direction as the velocity.
• 4.1.A.3 Momentum can be used to analyze collisions and explosions.
  • 4.1.A.3.i A collision is a model for an interaction where the forces exerted between the involved objects in the system are much larger than the net external force exerted on those objects during the interaction.
  • 4.1.A.3.ii As only the initial and final states of a collision are analyzed, the object model may be used to analyze collisions.
  • 4.1.A.3.iii An explosion is a model for an interaction in which forces internal to the system move objects within that system apart.

Learning Objective 4.2.A: Describe the impulse delivered to an object or system.

• 4.2.A.1 The rate of change of a system's momentum is equal to the net external force exerted on that system.
  • Equation: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt}$
• 4.2.A.2 Impulse is defined as the integral of a force exerted on an object or system over a time interval.
  • Equation: $\vec{J} = \displaystyle\int_{t_1}^{t_2} \vec{F}_{\text{net}}(t)\,dt$
• 4.2.A.3 Impulse is a vector quantity and has the same direction as the net force exerted on the system.
• 4.2.A.4 The impulse delivered to a system by a net external force is equal to the area under the curve of a graph of the net external force exerted on the system as a function of time.
• 4.2.A.5 The net external force exerted on a system is equal to the slope of a graph of the momentum of the system as a function of time.

Learning Objective 4.2.B: Describe the relationship between the impulse exerted on an object or system and the change in momentum of the object or system.

• 4.2.B.1 Change in momentum is the difference between a system's final momentum and its initial momentum.
  • Equation: $\Delta\vec{p} = \vec{p} - \vec{p}_0$
• 4.2.B.2 The impulse–momentum theorem relates the impulse delivered to an object and the object's change in momentum.
  • 4.2.B.2.i The impulse exerted on an object is equal to the object's change in momentum.
    • Equation: $\vec{J} = \displaystyle\int_{t_1}^{t_2} \vec{F}_{\text{net}}(t)\,dt = \Delta\vec{p}$
  • 4.2.B.2.ii Newton's second law of motion is a direct result of the impulse–momentum theorem applied to systems with constant mass.
    • Equation: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt} = m\dfrac{d\vec{v}}{dt} = m\vec{a}$
  • 4.2.B.2.iii The impulse–momentum theorem also describes the behavior of a system in which the velocity is constant but the mass changes with respect to time.
    • Equation: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt} = \dfrac{dm}{dt}\vec{v}$

Learning Objective 4.3.A: Describe the behavior of a system using conservation of linear momentum.

• 4.3.A.1 A collection of objects with individual momenta can be described as one system with one center-of-mass velocity.
  • 4.3.A.1.i For a collection of objects, the velocity of a system's center of mass can be calculated using the equation
    • Equation: $\vec{v}_{\text{cm}} = \dfrac{\sum \vec{p}_i}{\sum m_i} = \dfrac{\sum (m_i \vec{v}_i)}{\sum m_i}$
  • 4.3.A.1.ii The velocity of a system's center of mass is constant in the absence of a net external force.
• 4.3.A.2 The total momentum of a system is the sum of the momenta of the system's constituent parts.
• 4.3.A.3 In the absence of net external forces, any change to the momentum of an object within a system must be balanced by an equivalent and opposite change of momentum elsewhere within the system. Any change to the momentum of a system is due to a transfer of momentum between the system and its surroundings.
  • 4.3.A.3.i The impulse exerted by one object on a second object is equal and opposite to the impulse exerted by the second object on the first. This is a direct result of Newton's third law.
  • 4.3.A.3.ii A system may be selected so that the total momentum of that system is constant.
  • 4.3.A.3.iii If the total momentum of a system changes, that change will be equivalent to the impulse exerted on the system.
    • Equation: $\vec{J} = \Delta\vec{p}$
• 4.3.A.4 Correct application of conservation of momentum can be used to determine the velocity of a system immediately before and immediately after collisions or explosions.

Boundary statement: AP Physics C: Mechanics only expects students to quantitatively analyze collisions and interactions in one or two dimensions. Three-dimensional collisions may be analyzed qualitatively.

Learning Objective 4.3.B: Describe how the selection of a system determines whether the momentum of that system changes.

• 4.3.B.1 Momentum is conserved in all interactions.
• 4.3.B.2 If the net external force on the selected system is zero, the total momentum of the system is constant.
• 4.3.B.3 If the net external force on the selected system is nonzero, momentum is transferred between the system and the environment.

Learning Objective 4.4.A: Describe whether an interaction between objects is elastic or inelastic.

• 4.4.A.1 An elastic collision between objects is one in which the initial kinetic energy of the system is equal to the final kinetic energy of the system.
• 4.4.A.2 In an elastic collision, the final kinetic energies of each of the objects within the system may be different from their initial kinetic energies.
• 4.4.A.3 An inelastic collision between objects is one in which the total kinetic energy of the system decreases.
• 4.4.A.4 In an inelastic collision, some of the initial kinetic energy is not restored to kinetic energy but is transformed by nonconservative forces into other forms of energy.
• 4.4.A.5 In a perfectly inelastic collision, the objects stick together and move with the same velocity after the collision.