💥 Collisions

Types of Collisions

Collisions are classified based on what happens to kinetic energy:

| Type | Momentum Conserved? | Kinetic Energy Conserved? | What Happens | |------|-------------------|------------------------|--------------| | Elastic | ✅ Yes | ✅ Yes | Objects bounce off, no deformation | | Inelastic | ✅ Yes | ❌ No | Some KE lost to heat, sound, deformation | | Perfectly Inelastic | ✅ Yes | ❌ No (max loss) | Objects stick together |

💡 Key Point: Momentum is ALWAYS conserved in collisions (if system is isolated). Kinetic energy is only conserved in elastic collisions.

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Elastic Collisions

In an elastic collision:

• Momentum is conserved
• Kinetic energy is conserved
• Objects bounce apart
• No energy lost to heat, sound, or deformation

Examples:

• Colliding billiard balls (nearly elastic)
• Molecules in ideal gas
• Atomic particles (fundamental particles)

Conservation Equations (1D)

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

Kinetic Energy: $\frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2$

Two equations, two unknowns (if we know initial velocities and masses)

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Special Case: Equal Masses (Elastic)

If $m_1 = m_2$ and object 2 is initially at rest:

Before: Object 1 moving at $v$, object 2 at rest

After:

• Object 1 comes to rest ($v_{1f} = 0$)
• Object 2 moves with original velocity ($v_{2f} = v$)

They exchange velocities!

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Special Case: One Object Much Heavier

Light object hits heavy object at rest:

• Light object bounces back (reverses direction)
• Heavy object barely moves

Heavy object hits light object at rest:

• Heavy object continues forward (barely slows)
• Light object shoots forward at high speed

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Inelastic Collisions

In an inelastic collision:

• Momentum is conserved ✓
• Kinetic energy is NOT conserved ✗
• Some KE converted to other forms (heat, sound, deformation)
• Objects may or may not stick together

Examples:

• Car crash (crumple zones absorb energy)
• Dropped ball (loses energy on each bounce)
• Most real-world collisions

Energy Loss

$KE_{lost} = KE_i - KE_f$

$KE_{lost} = \left(\frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2\right) - \left(\frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2\right)$

This energy goes into:

• Heat (friction)
• Sound
• Deformation (crushing, bending)
• Internal energy

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Perfectly Inelastic Collisions

In a perfectly inelastic collision:

• Objects stick together after collision
• Move with same final velocity
• Maximum possible kinetic energy loss (for given initial conditions)
• Momentum still conserved!

Examples:

• Clay balls colliding
• Train cars coupling
• Tackle in football
• Meteor impact creating crater

Simplified Equation

Since objects stick together: $v_{1f} = v_{2f} = v_f$

Momentum conservation: $m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v_f$

$v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2}$

Only one unknown! Much simpler than elastic collisions.

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Coefficient of Restitution

The coefficient of restitution $e$ measures "bounciness":

$e = \frac{|v_{2f} - v_{1f}|}{|v_{1i} - v_{2i}|} = \frac{\text{relative speed after}}{\text{relative speed before}}$

Values:

• $e = 1$: Perfectly elastic (ideal bounce)
• $0 < e < 1$: Inelastic (some energy lost)
• $e = 0$: Perfectly inelastic (stick together, no bounce)

For a ball bouncing: $e = \sqrt{\frac{h_f}{h_i}}$

where $h_i$ is drop height, $h_f$ is bounce height.

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Problem-Solving Strategy

For Elastic Collisions:

1. Write momentum conservation: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$
2. Write energy conservation: $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$
3. Solve system of equations (algebra intensive!)
4. Check: KE should be equal before and after

For Perfectly Inelastic Collisions:

1. Use simplified equation: $m_1v_{1i} + m_2v_{2i} = (m_1 + m_2)v_f$
2. Solve for $v_f$
3. Optional: Calculate energy lost

Much simpler because objects stick together!

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⚠️ Common Mistakes

Mistake 1: Assuming KE is Always Conserved

❌ Wrong: KE is conserved in all collisions ✅ Right: KE is only conserved in ELASTIC collisions

Mistake 2: Perfectly Inelastic = No Motion

❌ Wrong: Perfectly inelastic means objects stop moving ✅ Right: Perfectly inelastic means objects stick together (they can still move!)

Mistake 3: Sign Errors in Momentum

Remember: velocities in opposite directions have opposite signs!

Mistake 4: Confusing Speed and Velocity

In coefficient of restitution, use relative speeds (magnitudes), but in momentum conservation use velocities (with signs).

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2D Collisions

For collisions in two dimensions:

Conserve momentum in EACH direction:

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

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

For elastic collisions, also conserve energy: $\frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2$

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

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Real-World Applications

Car Safety

Modern cars designed for inelastic collisions:

• Crumple zones deform (absorb energy)
• Reduces force on passengers
• Energy goes into crushing metal, not hurting people

Sports

Elastic collisions:

• Golf ball and club (nearly elastic for maximum energy transfer)
• Billiards (designed to be elastic)

Inelastic collisions:

• Football tackle (maximum energy dissipation)
• Boxing glove (padding increases collision time, decreases force)

Particle Physics

Particle accelerators create collisions:

• Analyze momentum and energy
• Discover new particles
• Test fundamental physics laws

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Comparing Collision Types

Same Initial Conditions

Consider object 1 (mass $m$, velocity $v$) hitting object 2 (mass $m$, at rest):

Elastic:

• Object 1 stops: $v_{1f} = 0$
• Object 2 moves: $v_{2f} = v$
• $KE_i = KE_f = \frac{1}{2}mv^2$
• No energy lost

Perfectly Inelastic:

• Both move together: $v_f = \frac{v}{2}$
• $KE_i = \frac{1}{2}mv^2$
• $KE_f = \frac{1}{2}(2m)\left(\frac{v}{2}\right)^2 = \frac{1}{4}mv^2$
• Half the kinetic energy lost!

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Explosions (Reverse Collisions)

An explosion is like a collision in reverse:

• Start: objects together (or at rest)
• End: objects flying apart

Still conserve momentum!

If explosion from rest: $m_1\vec{v}_1 + m_2\vec{v}_2 = 0$

Energy is ADDED to system (from chemical energy, spring, etc.)

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Key Formulas Summary

| Type | Momentum | Kinetic Energy | Formula | |------|----------|----------------|---------| | Elastic | Conserved | Conserved | Two conservation equations | | Inelastic | Conserved | Not conserved | Momentum equation only | | Perfectly Inelastic | Conserved | Not conserved (max loss) | $(m_1 + m_2)v_f = m_1v_{1i} + m_2v_{2i}$ | | Coefficient of restitution | - | - | $e = \frac{v_{separation}}{v_{approach}}$ |