Momentum and Collisions

Misconceptions

Key Concepts

Momentum

Momentum is mass times velocity:

• •p = mv
• •Vector quantity (has direction)
• •Unit: kg·m/s (no special name)
• •Total momentum of a system: p_total = Σmᵢvᵢ

Impulse

Impulse is the change in momentum caused by a force:

• •J = FΔt = Δp = p_f - p_i
• •Vector quantity
• •Unit: N·s = kg·m/s
• •Area under F-t graph equals impulse

Impulse-Momentum Theorem

The impulse equals the change in momentum:

• •FΔt = mΔv = mv_f - mv_i
• •Derived from Newton's second law: F = ma = m(Δv/Δt)

Conservation of Momentum

In an isolated system (no external forces), total momentum is conserved:

• •Σp_before = Σp_after
• •m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f

Types of Collisions

Elastic Collision:

• •Momentum is conserved
• •Kinetic energy is conserved
• •Objects bounce off each other
• •Examples: billiard balls, atomic collisions

Inelastic Collision:

• •Momentum is conserved
• •Kinetic energy is NOT conserved (some converts to heat, sound, deformation)
• •Most real collisions are inelastic

Perfectly Inelastic Collision:

• •Objects stick together after collision
• •Maximum KE loss (consistent with momentum conservation)
• •m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)v_f

Center of Mass

The point where total mass can be considered concentrated:

• •x_cm = (m₁x₁ + m₂x₂)/(m₁ + m₂)
• •Velocity of center of mass: v_cm = (m₁v₁ + m₂v₂)/(m₁ + m₂)
• •In isolated system, v_cm is constant

Equations

[1] p = mv (momentum)
[2] J = FΔt = Δp (impulse)
[3] m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f (conservation of momentum)
[4] (m₁ + m₂)v_f = m₁v₁ᵢ + m₂v₂ᵢ (perfectly inelastic)
[5] ½m₁v₁ᵢ² + ½m₂v₂ᵢ² = ½m₁v₁f² + ½m₂v₂f² (elastic collision, KE conserved)
[6] v₁f = ((m₁-m₂)/(m₁+m₂))v₁ᵢ + (2m₂/(m₁+m₂))v₂ᵢ (elastic collision formula)
[7] v_cm = (m₁v₁ + m₂v₂)/(m₁ + m₂) (center of mass velocity)

Worked Examples

Example 1: Impulse

Problem: A 0.15 kg baseball moving at 40 m/s is hit by a bat and leaves at 50 m/s in the opposite direction. What impulse did the bat deliver?

Solution:

1. •Define positive direction as initial ball direction
2. •Initial momentum: p_i = 0.15 × 40 = 6 kg·m/s
3. •Final momentum: p_f = 0.15 × (-50) = -7.5 kg·m/s
4. •Impulse: J = p_f - p_i = -7.5 - 6 = -13.5 kg·m/s (or 13.5 kg·m/s toward the pitcher)

Example 2: Perfectly Inelastic Collision

Problem: A 1000 kg car moving at 20 m/s collides with a 2000 kg truck at rest. They stick together. Find the final velocity.

Solution:

1. •Conservation of momentum: m₁v₁ + m₂v₂ = (m₁ + m₂)v_f
2. •1000(20) + 2000(0) = (1000 + 2000)v_f
3. •20000 = 3000v_f
4. •v_f = 6.67 m/s

Example 3: Elastic Collision (1D)

Problem: A 2 kg ball moving at 5 m/s collides elastically with a 3 kg ball at rest. Find both final velocities.

Solution:

1. •Using elastic collision formulas:
2. •v₁f = ((m₁-m₂)/(m₁+m₂))v₁ᵢ = ((2-3)/(2+3)) × 5 = (-1/5) × 5 = -1 m/s
3. •v₂f = (2m₁/(m₁+m₂))v₁ᵢ = (2×2/(2+3)) × 5 = (4/5) × 5 = 4 m/s
4. •Check: momentum and KE should be conserved

Explanation Patterns

1. •Define a coordinate system and positive direction
2. •Identify the system - what objects are involved?
3. •Check for external forces - if negligible, momentum is conserved
4. •Determine collision type: elastic (both p and KE conserved) or inelastic (only p conserved)
5. •Write conservation equations for each direction
6. •For 2D problems, use vector components
7. •Check your answer: total momentum before = total momentum after

Common Problem Types

1. •Impulse: FΔt = Δp, finding force from momentum change
2. •Inelastic collision: Objects stick together, find final velocity
3. •Elastic collision: Both momentum and KE conserved
4. •Recoil/explosion: Initially at rest, objects separate
5. •2D collision: Use vector components, conserve p_x and p_y separately
6. •Ballistic pendulum: Collision + energy conservation