Sample Notes: Momentum

AS Level Mathematics – Detailed Notes for Topic 4.3: Momentum

Definition and Nature of Momentum

Linear Momentum

• Definition: Linear momentum is the product of an object’s mass and velocity.
• Formula:
  p = m × v
  where:
  • p = momentum (kg·m/s)
  • m = mass (kg)
  • v = velocity (m/s)
• Vector Quantity:
  Momentum has both magnitude and direction.
  • If direction changes, the momentum changes even if speed is constant.

Understanding the Vector Nature

• Example:
  A 2 kg object moving at +4 m/s has momentum = 2 × 4 = +8 kg·m/s
  A 2 kg object moving at -4 m/s has momentum = 2 × -4 = -8 kg·m/s
• Momentum must always consider direction, especially in collision problems.

Law of Conservation of Linear Momentum

Principle:

• Total momentum before collision = Total momentum after collision
• Applies only when no external force acts on the system.

Formula:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Where:

• m₁, m₂ = masses of objects
• u₁, u₂ = initial velocities
• v₁, v₂ = final velocities

Types of Collisions

1. Elastic Collision

• Momentum conserved
• Kinetic energy also conserved
• Objects bounce off after impact

2. Inelastic Collision

• Momentum conserved
• Kinetic energy not conserved
• Some energy lost as heat, sound, etc.

3. Perfectly Inelastic Collision (Coalescing Bodies)

• Momentum conserved
• Bodies stick together after collision
• Final velocity is the same for both objects

Formula for Coalescing Bodies:

(m₁ + m₂)v = m₁u₁ + m₂u₂

Worked Examples

Example 1:

Two bodies collide:

• Object A: mass = 3 kg, velocity = 4 m/s
• Object B: mass = 2 kg, velocity = -3 m/s
  Find final velocity if they stick together:

Total initial momentum = (3 × 4) + (2 × -3) = 12 – 6 = 6 kg·m/s
Final mass = 3 + 2 = 5 kg
Final velocity = 6 / 5 = 1.2 m/s

Example 2:

A 2 kg trolley moving at 5 m/s collides with a 3 kg trolley at rest. After collision, they move separately.

Initial total momentum = (2 × 5) + (3 × 0) = 10 kg·m/s
If final velocities are v₁ and v₂:
2v₁ + 3v₂ = 10 (Use second equation from experiment or info given to solve)

Momentum vs Force

Newton’s Second Law (Momentum form):

F = Δp / Δt
Where:

• Δp = change in momentum
• Δt = time interval
• This version is especially useful in impact or collision questions

Key Exam Concepts and Tips

• Always choose one direction as positive (usually to the right).
• Use signs consistently (left = negative, right = positive).
• Include units: momentum in kg·m/s
• State if the collision is elastic or inelastic if relevant.
• If kinetic energy isn’t conserved, you must note it’s an inelastic collision.

Practice Concept Checks

• Can momentum be conserved but kinetic energy not?
  → Yes, in inelastic collisions.
• If two equal masses collide and stick together, what happens to the speed?
  → Final speed is average of initial velocities (if both move initially).

Important Constants & Units