4 Linear Momentum

Describe the linear momentum of an object or system.

• Linear momentum is defined by the equation pm v= .
• Momentum is a vector quantity and has the same direction as the velocity.
• Momentum can be used to analyze collisions and explosions.
  • 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.
  • ii. As only the initial and final states of a collision are analyzed, the object model may be used to analyze collisions.
  • iii. An explosion is a model for an interaction in which forces internal to the system move objects within that system apart.

Describe the impulse delivered to an object or system.

• The rate of change of a system’s momentum is equal to the net external force exerted on that system. Relevant equation: F dp dt net =  
• Impulse is defined as the integral of a force exerted on an object or system over a time interval. Relevant equation:
• Impulse is a vector quantity and has the same direction as the net force exerted on the system.
• 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.
• 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. TOPIC 4.2 Change in Momentum and Impulse

Describe the relationship between the impulse exerted on an object or system and the change in momentum of the object or system.

• Change in momentum is the difference between a system’s final momentum and its initial momentum. Relevant equation:
• The impulse–momentum theorem relates the impulse delivered to an object and the object’s change in momentum.
  • i. The impulse exerted on an object is equal to the object’s change in momentum. Relevant equation:
  • ii. Newton’s second law of motion is a direct result of the impulse–momentum theorem applied to systems with constant mass. F dp dt m dv dt manet == =    
  • 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. F dp dt dm dt vnet ==   

Describe the behavior of a system using conservation of linear momentum.

• A collection of objects with individual momenta can be described as one system with one center-of-mass velocity.
  • i. For a collection of objects, the velocity of a system’s center of mass can be calculated using the equation
  • ii. The velocity of a system’s center of mass is constant in the absence of a net external force.
• The total momentum of a system is the sum of the momenta of the system’s constituent parts.
• 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.
  • 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. TOPIC 4.3 Conservation of Linear Momentum
  • ii. A system may be selected so that the total momentum of that system is constant.
  • iii. If the total momentum of a system changes, that change will be equivalent to the impulse exerted on the system. Relevant equation:
• Correct application of conservation of momentum can be used to determine the velocity of a system immediately before and immediately after collisions or explosions.

Describe how the selection of a system determines whether the momentum of that system changes.

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

Describe whether an interaction between objects is elastic or inelastic.

• 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.
• In an elastic collision, the final kinetic energies of each of the objects within the system may be different from their initial kinetic energies.
• An inelastic collision between objects is one in which the total kinetic energy of the system decreases.
• 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.
• In a perfectly inelastic collision, the objects stick together and move with the same velocity after the collision. TOPIC 4.4 Elastic and Inelastic Collisions Topic 5.1 R otational Kinematics