8.3 Elastic and Inelastic Collisions

Elastic and Inelastic Collisions

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Elastic vs inelastic collisions

Every collision conserves momentum. The difference between collision types comes down to what happens to kinetic energy.

Elastic collisions conserve both momentum and kinetic energy. The total kinetic energy before and after the collision stays the same. Objects bounce off each other without any energy lost to heat, sound, or deformation. True elastic collisions are rare in everyday life but are commonly observed between atomic and subatomic particles. Newton's cradle is a close approximation at the macroscopic scale.

Inelastic collisions conserve momentum but not kinetic energy. Some kinetic energy gets converted into other forms like heat, sound, or permanent deformation. There are two subcategories:

• Perfectly inelastic collisions: The objects stick together after colliding and move as one combined mass. This is the maximum possible kinetic energy loss for a given momentum exchange. Think of two lumps of clay merging on impact, or two football players tackling and moving together.
• Partially inelastic collisions: The objects separate after colliding but still lose some kinetic energy. This is the most common type in real life. Car bumpers crumpling during a fender-bender is a good example: the cars bounce apart, but the deformation absorbs energy.

Conservation of momentum in collisions

The law of conservation of momentum states that the total momentum of a closed system (no external net forces) remains constant. For a two-object collision:

$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$

• $m_1$, $m_2$ are the masses of the two objects
• $v_1$, $v_2$ are the initial velocities (before collision)
• $v_1'$, $v_2'$ are the final velocities (after collision)

One-dimensional collisions are the simplest case. Both objects move along a single line, so you only need one momentum equation. Be careful with signs: pick a positive direction and stick with it. An object moving left when you've defined right as positive has a negative velocity.

Two-dimensional collisions (like billiard balls hitting at an angle) require you to work with components:

1. Choose a coordinate system (x and y axes).

2. Resolve each object's velocity into x and y components before and after the collision.

3. Apply conservation of momentum separately for x and y:

   • x: $m_1v_{1x} + m_2v_{2x} = m_1v_{1x}' + m_2v_{2x}'$
   • y: $m_1v_{1y} + m_2v_{2y} = m_1v_{1y}' + m_2v_{2y}'$

4. Solve the two equations for the unknowns, then recombine the components to find the final speed and direction of each object using the Pythagorean theorem and inverse tangent.

Momentum and kinetic energy relationships

Momentum and kinetic energy both depend on mass and velocity, but they scale differently:

• Momentum: $p = mv$ (linear in velocity)
• Kinetic energy: $KE = \frac{1}{2}mv^2$ (quadratic in velocity)

This difference is why two quantities can behave independently in collisions. Doubling an object's speed doubles its momentum but quadruples its kinetic energy.

In elastic collisions, both are conserved simultaneously:

$\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2$

Having two conservation equations (momentum and kinetic energy) means you can solve for two unknowns, which is exactly what you need for a two-object elastic collision.

In inelastic collisions, kinetic energy decreases:

$\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 > \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2$

The "missing" kinetic energy went into deformation, heat, sound, etc.

For perfectly inelastic collisions, the two objects share a single final velocity, so the momentum equation simplifies to:

$v_f = \frac{m_1v_1 + m_2v_2}{m_1 + m_2}$

This is one of the most useful equations in this unit. For example, if a 2 kg cart moving at 3 m/s collides with and sticks to a stationary 4 kg cart: $v_f = \frac{(2)(3) + (4)(0)}{2 + 4} = 1 \text{ m/s}$. You can then calculate how much kinetic energy was lost by comparing $KE$ before and after.

The work-energy theorem connects to collisions as well: the net work done on an object equals its change in kinetic energy. During a collision, internal forces do work that can transfer or transform kinetic energy.

Reference frames and relative motion

The center of mass (COM) frame is a reference frame where the total momentum of the system is zero. Analyzing a collision in this frame simplifies the math because the two objects always approach each other with equal and opposite momenta. After solving in the COM frame, you can transform back to the lab frame.

Relative velocity between two objects determines collision intensity. For elastic collisions, there's a useful result: the relative speed of approach before the collision equals the relative speed of separation after. If two objects approach each other at a combined 5 m/s, they'll separate at a combined 5 m/s in a perfectly elastic collision. In inelastic collisions, the separation speed is always less than the approach speed.