Momentum,+Impulse+and+Conservation+of+Momentum

Chapter 7: Impulse and Momentum

Impulse-Momentum Theorem

• The concepts of impulse and momentum help in understanding collisions (e.g., a soccer ball and a player's head)

• Impulse-Momentum Theorem relates to systems where the force acting on an object is not constant

• Example: In a baseball being hit, the force exerted by the bat varies over time, starting at zero, peaking, and returning to zero once the contact is broken

• The average force experienced during this short time interval affects the motion of the object.

Key Concepts

• Impulse (J):

  • Defined as the product of average force (F) and the time interval (At) during which the force acts.

  • Formula:

    • [ J = F \Delta t ]

  • Impulse is a vector quantity with the same direction as the average force.

  • SI Unit: Newton-second (N·s)

• Linear Momentum (p):

  • Defined as the product of mass (m) and velocity (v).

  • Formula:

    • [ p = mv ]

  • Linear momentum is also a vector pointing in the same direction as velocity.

  • SI Unit: Kilogram meter/second (kg·m/s)

Relationship Between Impulse and Momentum

• The impulse-momentum theorem states that the impulse of a force acting on an object is equal to the change in momentum of the object:

  • [ J = \Delta p = p_{final} - p_{initial} ]

Applications of Theorem

• Used extensively in analyzing collisions

• Example: Hitting a baseball requires understanding both the magnitude of the force and the duration of contact between bat and ball.

Conservation of Linear Momentum

• Total linear momentum of an isolated system is conserved when external forces sum to zero:

  • [ \Delta P = P_{final} - P_{initial} = 0 ]

• Impulse-Momentum Theorem leads to this conservation

  • Significance in collisions: the initial and final momentum of the system remains consistent in the absence of net external forces

Types of Collisions

• Elastic Collision:

  • Total kinetic energy is conserved.

• Inelastic Collision:

  • Total kinetic energy of the system is not conserved.

  • If the objects stick together, it's called a completely inelastic collision.

Center of Mass

• Center of mass is the average location of the total mass in a system

• Important in analyzing motion and momentum of systems with multiple particles.

  • Formula to find the center of mass for two particles:

    • [ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} ]

• Velocity of the center of mass:

  • [ v_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} ]

• During collisions or interactions, if total momentum remains constant, the velocity of the center of mass does as well.

Problem Solving Strategy

1. Identify the objects in the system.

2. Determine forces acting on the system (internal vs. external).

3. Confirm whether the system is isolated (external forces sum to zero).

4. Apply conservation of momentum to find unknowns.