Crash Course — Unit 4: Linear Momentum

• Linear Momentum (p)
  €” The product of an object's mass and velocity (p = mv). It's a vector, meaning its direction is critical.
• Impulse (J)
  €” The change in momentum, caused by a net force acting over a period of time (J = F_netΔt). It's the "kick" an object receives.
• Impulse-Momentum Theorem
  €” The direct link between force and momentum: the impulse delivered to an object equals its change in momentum (J = Δp).
• System
  €” The collection of objects you choose to analyze. Defining your system correctly is the first step in any conservation problem.
• Internal vs. External Forces
  €” Internal forces are between objects inside your system (like two skaters pushing off each other). External forces come from outside the system (like gravity from the Earth or friction from the floor).
• Conservation of Linear Momentum
  €” If the net external force on a system is zero, its total momentum cannot change. The total momentum before an interaction equals the total momentum after.
• Collision
  €” Any interaction where objects exert forces on each other over a short time. Momentum is conserved for the system.
• Inelastic Collision
  €” A collision where kinetic energy is not conserved (it's lost to heat, sound, or deformation). Most real-world collisions are inelastic.
• Perfectly Inelastic Collision
  €” A type of inelastic collision where the objects stick together after impact, moving with a single final velocity.
• Elastic Collision
  €” A special, idealized collision where kinetic energy is conserved, along with momentum. Think of perfect, silent billiard balls.

Key Formulas / Terms

• p = mv — Definition of momentum. Remember that v is a vector, so p is too!
• J = F_netΔt = Δp — The Impulse-Momentum Theorem. This is on your formula sheet and connects everything in the unit.
• Area under Force-vs-Time graph = Impulse (Δp) — A common graphical application.
• Σp_initial = Σp_final — The Law of Conservation of Momentum. This is your starting point for all collision and explosion problems.
• m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁₟ + m₂v₂₟ — The general template for a two-object collision.
• m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)v₟ — The specific template for a perfectly inelastic (stick-together) collision.
• ΣK_initial = ΣK_final — Only use this for elastic collisions. For all other collisions, K_final < K_initial.

Exam Traps

• Trap
  Forgetting momentum is a vector. The AP exam loves setting up 1D collisions where objects move in opposite directions. If you treat all speeds as positive, you'll get the wrong answer.

  • Counter: At the start of every problem, explicitly define a positive direction (e.g., "to the right is positive"). Any velocity pointing left must have a negative sign.
• Trap
  Confusing conservation of momentum and conservation of kinetic energy. Students assume if one is conserved, the other must be.

  • Counter: Drill this into your head: In an isolated system, momentum is always conserved in a collision. Kinetic energy is only conserved in the special case of an elastic collision.
• Trap
  Incorrectly defining the system. You might analyze a single bouncing ball and assume its momentum is conserved. It's not, because the floor exerts a massive external force on it.

  • Counter: Before you write p_initial = p_final, ask: "Are there any significant external forces on my chosen system?" If yes (like friction, or the floor), momentum is not conserved. To use conservation, you must expand your system to include the source of that force (e.g., the ball + Earth system).
• Trap
  Reading a Force-vs-Time graph incorrectly. Students see a question asking for impulse and just read the peak value of the force from the y-axis.

  • Counter: Impulse (J or Δp) is the area under the Force-vs-Time graph. If the shape is a triangle, the area is ½ * base * height.
• Trap
  Treating "explosions" differently from collisions. A problem where a cannon fires a cannonball, or two ice skaters push off from rest, can feel confusing.

  • Counter: An explosion is just a collision in reverse. Use conservation of momentum. If the system starts from rest, the total initial momentum is zero. The final momenta of all the pieces must add up (as vectors) to zero.