Chapter 5: Impulse and Momentum

Problem Solver's Guide

Your strategy guide for tackling any impulse and momentum problem.

1. Impulse from Force-Time Graph

Key Idea:

The impulse is the area under the Force vs. Time curve.

$$ I = \int F \,dt = \text{Area} $$

Strategy:

  1. Calculate the geometric area of the graph (rectangle, triangle, etc.).
  2. Set the calculated area equal to Impulse (I).
  3. Use \( I = \Delta p \) to find the change in velocity: \( v_f = v_i + \frac{I}{m} \).

2. Impulse-Momentum Theorem

Key Idea:

Directly relates force, time, and change in momentum.

$$ F_{avg}\Delta t = \Delta p = m(v_f - v_i) $$

Strategy:

  1. Identify all known variables (mass, velocities, time, force).
  2. Plug the knowns into the theorem's equation.
  3. Solve algebraically for the single unknown variable.

3. Conservation of Momentum

Key Idea:

In an isolated system, total momentum before an event equals total momentum after.

$$ \sum p_i = \sum p_f $$

Strategy:

  1. Draw "before" and "after" diagrams.
  2. Sum all momentum vectors before the event.
  3. Sum all momentum vectors after the event.
  4. Set them equal and solve for the unknown(s).

4. Perfectly Inelastic Collision

Key Idea:

Objects collide and stick together, moving as one mass.

$$ m_1v_{1i} + m_2v_{2i} = (m_1 + m_2)v_f $$

Strategy:

  1. Plug in known masses and initial velocities.
  2. Solve for the common final velocity, \(v_f\).
  3. If asked, calculate \( \Delta KE = KE_f - KE_i \) to find energy loss.

5. Elastic Collision (1D)

Key Idea:

Objects bounce off each other. Both momentum and kinetic energy are conserved.

$$ v_{1i} - v_{2i} = -(v_{1f} - v_{2f}) $$

Strategy:

  1. Set up two equations: momentum conservation and the relative velocity equation above.
  2. Solve the system of two equations for the two unknowns, \(v_{1f}\) and \(v_{2f}\).

6. 2D Collisions

Key Idea:

Momentum is conserved independently in both the x and y directions.

$$ \sum p_{x,i} = \sum p_{x,f} \quad \text{and} \quad \sum p_{y,i} = \sum p_{y,f} $$

Strategy:

  1. Break all initial velocity vectors into x and y components.
  2. Set up a momentum conservation equation for each axis.
  3. Solve the resulting system of two equations for the unknowns.

7. Explosions (Recoil)

Key Idea:

An object breaks apart. Initial momentum is often zero.

Diagram of a cannon firing, illustrating recoil
$$ 0 = m_1v_{1f} + m_2v_{2f} + ... $$

Strategy:

  1. The total momentum before the explosion is the same as after (often zero).
  2. Sum the momenta of all pieces after the explosion.
  3. Set the sum equal to the initial momentum and solve.

8. Kinetic Energy Loss

Key Idea:

Finding the energy converted to heat, sound, or deformation in a collision.

$$ \Delta KE = KE_f - KE_i $$

Strategy:

  1. Calculate total system kinetic energy before the collision (\(KE_i\)).
  2. Calculate total system kinetic energy after the collision (\(KE_f\)).
  3. Find the difference. A negative result means energy was lost.