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MECHANICS FOR PP4

M1 TOPIC 3 : Momentum

Momentum - CIE Mechanics 1

📚 CIE Mechanics 1 - Topic 4.3: Momentum and Conservation of Linear Momentum

Definition of Linear Momentum

Linear Momentum (p) is defined as the product of an object's mass and its velocity.

p = mv

Where:
• p = momentum (kg m/s)
• m = mass (kg)
• v = velocity (m/s)

Vector Nature: Momentum is a vector quantity - it has both magnitude and direction. The direction of momentum is the same as the direction of velocity.

Example 1: Calculating Momentum

A car of mass 1200 kg travels east at 25 m/s. Calculate its momentum.

Solution:

p = mv = 1200 × 25 = 30,000 kg m/s east

Conservation of Linear Momentum

Principle of Conservation of Momentum: In a closed system with no external forces, the total momentum before a collision equals the total momentum after the collision.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

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

Important: Velocities are vectors - direction matters! Use positive and negative signs for opposite directions.

Types of Collisions

1. Direct Impact - Bodies Coalesce (Stick Together)

m₁u₁ + m₂u₂ = (m₁ + m₂)v

Example 2: Coalescing Collision

A car of mass 1000 kg moving at 20 m/s collides with a stationary car of mass 1500 kg. If they stick together, find their common velocity after collision.

Step 1: Identify values
m₁ = 1000 kg, u₁ = 20 m/s
m₂ = 1500 kg, u₂ = 0 m/s

Step 2: Apply conservation
m₁u₁ + m₂u₂ = (m₁ + m₂)v
(1000 × 20) + (1500 × 0) = (1000 + 1500)v

Step 3: Solve
20,000 = 2500v
v = 8 m/s

Answer: 8 m/s in original direction

2. Direct Impact - Bodies Separate

Example 3: Separating Collision

A white snooker ball (mass 0.17 kg) moving at 2 m/s hits a stationary red ball (mass 0.17 kg). After collision, the white ball continues at 0.5 m/s. Find the red ball's velocity.

Step 1: Identify values
m₁ = 0.17 kg, u₁ = 2 m/s, v₁ = 0.5 m/s
m₂ = 0.17 kg, u₂ = 0 m/s, v₂ = ?

Step 2: Apply conservation
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
(0.17 × 2) + (0.17 × 0) = (0.17 × 0.5) + (0.17 × v₂)

Step 3: Solve
0.34 = 0.085 + 0.17v₂
v₂ = 1.5 m/s

Answer: 1.5 m/s in same direction

Working with Opposite Directions

Sign Convention: Choose a positive direction. Velocities in opposite direction are negative.

Example 4: Head-on Collision

Two trolleys approach each other. Trolley A (2 kg) moves right at 3 m/s, trolley B (3 kg) moves left at 2 m/s. They collide and stick together. Find final velocity.

Step 1: Choose direction
Right = positive, Left = negative
m₁ = 2 kg, u₁ = +3 m/s
m₂ = 3 kg, u₂ = -2 m/s

Step 2: Apply conservation
m₁u₁ + m₂u₂ = (m₁ + m₂)v
(2 × 3) + (3 × -2) = (2 + 3)v

Step 3: Solve
6 + (-6) = 5v
v = 0 m/s

Answer: The combined trolleys come to rest.

Problem-Solving Strategy

1. Draw diagram - Show before/after collision

2. Choose positive direction - Define clearly

3. List known values - With correct signs

4. Apply conservation - Total momentum before = after

5. Solve equation - Find unknown velocity

6. Interpret result - Include direction

Syllabus Limitations:
• Motion in one dimension only
• No impulse required
• No coefficient of restitution
• Focus on direct impact problems

Common Mistakes

Forgetting direction: Momentum is a vector!

Incorrect signs: Be consistent with positive direction

Units mismatch: Mass in kg, velocity in m/s

Practice Problems

Problem 1

A 5 kg object moving at 4 m/s collides with a stationary 3 kg object. If they stick together, find their common velocity.

Problem 2

Two ice skaters push off from each other. Skater A (60 kg) moves left at 2 m/s. If skater B (75 kg) moves right, find skater B's velocity.

Exam Tip: Always show working clearly. Use diagrams and state sign convention.