Unit 5 Overview: Momentum

Momentum is a cornerstone concept in physics, essential for analyzing motion, collisions, and force interactions. This unit accounts for 12-18% of the AP exam and spans approximately 14-17 class periods.

Key Concepts of Momentum

1. Momentum ($p$):

   • Formula: $$p = mv$$
     ($p$: momentum, $m$: mass, $v$: velocity).
   • Vector quantity (has both magnitude and direction).

2. Law of Conservation of Momentum:

   • Total momentum in an isolated system remains constant.
   • Key Application: In collisions, the momentum before equals momentum after:
     $$\sum p_{\text{initial}} = \sum p_{\text{final}}$$

3. Impulse ($J$):

   • Change in momentum over time.
   • Formula: $$J = \Delta p = F \Delta t$$($F$: force, $\Delta t$: time).

4. Collisions:

   • Elastic: Kinetic energy and momentum are conserved.
   • Inelastic: Momentum conserved; kinetic energy not conserved.

5. Work-Energy Theorem:

   • Work done on an object equals its change in kinetic energy.

5.1 Momentum and Impulse

Momentum measures an object’s “amount of motion.” It’s proportional to mass and velocity.

Impulse and Momentum Change

Impulse quantifies the force applied over time to change momentum:

$$J = F \Delta t = \Delta p$$

• Applications:
  • Catching a ball: Prolonging the time of impact reduces the force needed.
  • Car airbags: Increase $\Delta t$ to decrease $F$.

5.2 Representations of Momentum

Momentum can be visualized in multiple ways:

1. Vector Notation:

   • Represented as an arrow; length indicates magnitude, direction shows velocity.

2. Component Form:

   • Break momentum into $x$– and $y$-components: $$p_x = mv_x, \, p_y = mv_y$$

3. Graphical Representation:

   • Use coordinate axes for momentum in $x$– and $y$-directions.

4. Momentum Diagrams:

   • Arrows depict momentum before and after collisions, showing conservation visually.

5.3 Open and Closed Systems: Momentum

• Closed Systems:

  • No exchange of matter or energy with surroundings.
  • Momentum conserved.

• Open Systems:

  • Exchange of matter or energy occurs.
  • Momentum may change due to external forces (e.g., wall exerting force on a ball).

5.4 Conservation of Linear Momentum

Linear momentum conservation applies to closed systems:

$$\sum p_{\text{initial}} = \sum p_{\text{final}}$$

Applications:

1. Collisions:

   • Elastic: Momentum and kinetic energy conserved.
   • Inelastic: Momentum conserved; objects may stick together.

2. Rocket Propulsion:

   • Total momentum (rocket + exhaust gases) remains constant as rocket accelerates.

Real-World Applications of Momentum

1. Car Crashes:

   • Crumple zones increase $\Delta t$, reducing force on passengers.

2. Sports:

   • Applying impulse to control ball speed and direction in tennis or soccer.

3. Space Exploration:

   • Rockets eject exhaust gases to propel forward using conservation of momentum.

Practice Problem

Scenario:

A 3 kg ball moving at 4 m/s collides with a stationary 2 kg ball. After the collision, the 3 kg ball moves at 2 m/s. Find the velocity of the 2 kg ball.

Solution:

1. Initial Momentum:

   $$p_{\text{initial}} = m_1v_1 + m_2v_2 = (3)(4) + (2)(0) = 12 \, \text{kg·m/s}$$
   

2. Final Momentum:

   $$p_{\text{final}} = (3)(2) + (2)v_2$$

3. Conservation of Momentum:

   $$p_{\text{initial}} = p_{\text{final}}$$ $$12 = 6 + 2v_2$$ $$v_2 = 3 \, \text{m/s}$$
   

The velocity of the 2 kg ball after the collision is $3 \, \text{m/s}$.