$m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f}$

💡 Fundamental Law: This is one of the most important conservation laws in physics! It applies to all interactions, from colliding billiard balls to exploding stars.

Isolated System

A system is isolated when:

• No external forces act on the system
• OR external forces sum to zero
• OR external forces are negligible compared to internal forces

Examples of isolated systems:

• Two ice skaters pushing apart (on frictionless ice)
• Collision between two cars (during brief collision, friction negligible)
• Explosion of fireworks (no external forces during explosion)

Not isolated:

• Ball rolling on rough surface (friction is external force)
• Rocket launching (thrust is internal, but gravity/air resistance are external)

Why is Momentum Conserved?

Derivation from Newton's Laws

For two objects interacting:

By Newton's 3rd Law: $\vec{F}_{12} = -\vec{F}_{21}$

$\frac{d\vec{p}_1}{dt} = -\frac{d\vec{p}_2}{dt}$

$\frac{d\vec{p}_1}{dt} + \frac{d\vec{p}_2}{dt} = 0$

$\frac{d(\vec{p}_1 + \vec{p}_2)}{dt} = 0$

Therefore: $\vec{p}_1 + \vec{p}_2 =$ constant ✓

Internal forces come in action-reaction pairs that cancel!

One-Dimensional Collisions

For motion along a line (use + and - for direction):

$m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

Problem-Solving Steps:

1. Choose positive direction (usually direction of motion)
2. List known values with proper signs
3. Write conservation equation
4. Solve for unknown

Two-Dimensional Collisions

Momentum is conserved in each direction independently:

x-direction: $m_1 v_{1ix} + m_2 v_{2ix} = m_1 v_{1fx} + m_2 v_{2fx}$

y-direction: $m_1 v_{1iy} + m_2 v_{2iy} = m_1 v_{1fy} + m_2 v_{2fy}$

Need to use components: $v_x = v\cos\theta$, $v_y = v\sin\theta$

Explosions and Recoil

In an explosion, objects initially at rest fly apart:

Before: $p_i = 0$ (at rest)

After: $p_f = m_1\vec{v}_1 + m_2\vec{v}_2 = 0$

Therefore: $m_1\vec{v}_1 = -m_2\vec{v}_2$

Objects move in opposite directions with momenta equal in magnitude.

Examples:

• Rifle recoil: Bullet goes forward, rifle goes backward
• Rocket propulsion: Exhaust goes backward, rocket goes forward
• Ice skaters pushing apart: Equal and opposite momenta

Center of Mass

The center of mass of an isolated system moves at constant velocity:

$\vec{v}_{cm} = \frac{m_1\vec{v}_1 + m_2\vec{v}_2}{m_1 + m_2} = \frac{\vec{p}_{total}}{m_{total}}$

If system is isolated, $\vec{p}_{total}$ is constant, so $\vec{v}_{cm}$ is constant!

Example: Two skaters pushing apart

• Each skater accelerates
• But center of mass continues at same velocity (or stays at rest)

When is Momentum NOT Conserved?

Momentum is NOT conserved when:

• External forces act on system
• Friction is significant
• System is not isolated

However: Even with external forces, conservation can apply during brief interactions where internal forces dominate.

Example: Car collision

• Friction acts on system (external)
• But during 0.1 s collision, collision forces >> friction
• Momentum approximately conserved during collision

⚠️ Common Mistakes

Mistake 1: Forgetting Vector Nature

In 2D problems, must conserve momentum in EACH direction separately.

Mistake 2: Wrong Signs

Velocities in opposite directions have opposite signs!

• Object moving right: $v > 0$
• Object moving left: $v < 0$

Mistake 3: Including External Forces

Don't apply conservation if significant external forces act on system!

Mistake 4: Confusing Before and After

Make sure you clearly identify which velocities are initial ($v_i$) and which are final ($v_f$).

Momentum vs. Energy Conservation

In inelastic collisions:

• Momentum IS conserved ✓
• Kinetic energy is NOT conserved (some converts to heat, sound, deformation)

Problem-Solving Strategy

For Collision Problems:

1. Identify the system (what objects are involved?)
2. Check if isolated (are external forces negligible?)
3. Choose coordinate system (which direction is positive?)
4. List knowns and unknowns
   • Before collision: $m_1, v_{1i}, m_2, v_{2i}$
   • After collision: $v_{1f}, v_{2f}$ (unknowns?)
5. Apply conservation of momentum
   • 1D: One equation
   • 2D: Two equations (x and y)
6. Solve algebraically
7. Check your answer (reasonable magnitude? correct direction?)

Applications

Rocket Propulsion

Momentum of rocket + exhaust is conserved: $m_{rocket}v_{rocket} + m_{exhaust}v_{exhaust} = 0$

Exhaust goes backward → rocket goes forward

Particle Physics

When particles collide or decay, total momentum is always conserved. Used to detect invisible particles!

Asteroid Defense

To deflect asteroid:

• Explosion changes asteroid's momentum
• Conservation tells us required impulse

Traffic Accidents

Forensics use conservation of momentum to determine pre-collision speeds from post-collision debris patterns.

Key Formulas Summary

Find: Final velocity $v_f$

Step 1: Identify the system

System = car + truck (isolated during collision)

Step 2: Calculate initial momentum

$p_i = m_1 v_{1i} + m_2 v_{2i}$

$p_i = (1000)(20) + (2000)(0)$

$p_i = 20,000 \text{ kg·m/s}$

Step 3: Calculate final momentum

After sticking together, combined mass moves with velocity $v_f$:

$p_f = (m_1 + m_2)v_f$

$p_f = (1000 + 2000)v_f = 3000v_f$

Step 4: Apply conservation of momentum

$p_i = p_f$

$20,000 = 3000v_f$

$v_f = \frac{20,000}{3000} = 6.67 \text{ m/s}$

Answer: The velocity immediately after collision is 6.67 m/s in the direction the car was traveling.

Check: Final velocity (6.67 m/s) is less than initial car velocity (20 m/s), which makes sense since the car must slow down when it hits the truck. ✓