4.4 Elastic and Inelastic Collisions

Elastic vs Inelastic Interactions

Collisions between objects can be categorized based on how energy is conserved during the interaction. This fundamental distinction helps physicists analyze and predict the outcomes of various collision scenarios.

• Elastic collisions maintain the system's total kinetic energy, like pool balls bouncing off each other 🎱
• Inelastic collisions result in some energy converting to other forms like heat or sound 🔥
• Real-world collisions typically fall somewhere on the spectrum between perfectly elastic and perfectly inelastic

In everyday life, most collisions we observe are somewhat inelastic, with billiard ball collisions being close to elastic and car crashes being highly inelastic.

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Elastic Collision Energy Conservation

In an elastic collision, kinetic energy is completely preserved, allowing objects to bounce away from each other with no energy "lost" to other forms.

The mathematical expression of energy conservation in elastic collisions is: $KE_{initial} = KE_{final}$

This means:

• The sum of kinetic energies before collision equals the sum after collision
• Both momentum and energy conservation equations are needed to solve for unknown velocities

For a two-object collision, we can write: $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$

Individual Object Energy Changes

During elastic collisions, even though the system's total kinetic energy stays the same, each individual object's kinetic energy may change. Energy transfers between objects in predictable ways that depend on their relative masses and initial velocities.

When objects of different masses collide elastically:

• A heavier object transfers only a portion of its energy to a lighter one
• A lighter object can bounce off a heavier one with significantly increased speed
• When identical masses collide head-on, they essentially exchange velocities

For example, when a moving ball elastically strikes an identical stationary ball head-on, the first ball stops completely while the second moves away with the initial velocity of the first.

Inelastic Collision Energy Decrease

In inelastic collisions, some kinetic energy converts to other forms, resulting in a measurable decrease in the system's total kinetic energy.

$KE_{initial} > KE_{final}$

The "missing" kinetic energy doesn't violate energy conservation but transforms into:

• Thermal energy through friction and deformation
• Sound energy as vibrations travel through air
• Potential energy stored in deformed materials
• Other forms like light (in extreme high-energy collisions)

Car safety features intentionally make collisions more inelastic, allowing crumple zones to absorb energy that would otherwise be transferred to passengers.

Energy Transformation in Collisions

The energy that appears to be "lost" in inelastic collisions actually undergoes transformation into other forms, following the law of energy conservation. In these collisions, nonconservative forces (like friction and deformation forces) do work on the objects, converting kinetic energy into other energy types.

During a collision, kinetic energy may transform through several mechanisms:

• Friction between surfaces generates heat through molecular vibrations
• Deformation of materials stores energy temporarily (elastic) or permanently (plastic)
• Vibrations in the colliding objects produce sound waves that propagate outward
• Internal friction within materials converts mechanical energy to thermal energy

The extent of these transformations depends on material properties, collision speed, and geometry. Softer materials generally convert more kinetic energy to other forms than harder materials.

Perfectly Inelastic Collisions

The extreme case of inelastic collisions occurs when objects stick together after impact, resulting in maximum kinetic energy loss while still preserving momentum.

In perfectly inelastic collisions:

• Objects move together with a common final velocity after collision
• The final velocity can be calculated using only conservation of momentum: $v_f = \frac{m_1v_1 + m_2v_2}{m_1 + m_2}$
• In a perfectly inelastic collision, the amount of kinetic energy lost depends on the masses and initial velocities of the objects. What always defines the collision is that the objects stick together and move with the same final velocity, while momentum is conserved and the system's total kinetic energy decreases.
• Real-world examples include bullets embedding in targets, vehicles crashing and entangling, and objects sticking together with adhesives

The kinetic energy lost in a perfectly inelastic collision can be calculated by finding the difference between initial and final kinetic energies of the system.

Practice Problem 1: Elastic Collisions

Solution

To solve this problem, we need to apply both conservation of momentum and conservation of kinetic energy:

1. Conservation of momentum: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$ $(2.0 \text{ kg})(3.0 \text{ m/s}) + (1.0 \text{ kg})(0 \text{ m/s}) = (2.0 \text{ kg})v_{1f} + (1.0 \text{ kg})v_{2f}$ $6.0 \text{ kg}\cdot\text{m/s} = 2.0v_{1f} + 1.0v_{2f}$

2. Conservation of kinetic energy: $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$ $\frac{1}{2}(2.0)(3.0)^2 + \frac{1}{2}(1.0)(0)^2 = \frac{1}{2}(2.0)v_{1f}^2 + \frac{1}{2}(1.0)v_{2f}^2$ $9.0 \text{ J} = 1.0v_{1f}^2 + 0.5v_{2f}^2$

Solving these equations: $v_{1f} = 1.0 \text{ m/s}$ $v_{2f} = 4.0 \text{ m/s}$

After the collision, the 2.0 kg ball continues moving in the same direction at 1.0 m/s, while the 1.0 kg ball moves at 4.0 m/s in the same direction.

Practice Problem 2: Perfectly Inelastic Collisions

Solution

For a perfectly inelastic collision, we can use conservation of momentum to find the final velocity:

1. Conservation of momentum: $m_1v_{1i} + m_2v_{2i} = (m_1 + m_2)v_f$ $(1500 \text{ kg})(20 \text{ m/s}) + (2500 \text{ kg})(0 \text{ m/s}) = (1500 \text{ kg} + 2500 \text{ kg})v_f$ $30,000 \text{ kg}\cdot\text{m/s} = 4000 \text{ kg} \times v_f$ $v_f = 7.5 \text{ m/s}$

2. Calculating kinetic energy loss: Initial kinetic energy: $KE_i = \frac{1}{2}m_1v_{1i}^2 = \frac{1}{2}(1500)(20)^2 = 300,000 \text{ J}$ Final kinetic energy: $KE_f = \frac{1}{2}(m_1+m_2)v_f^2 = \frac{1}{2}(4000)(7.5)^2 = 112,500 \text{ J}$ Energy lost: $KE_i - KE_f = 300,000 - 112,500 = 187,500 \text{ J}$

The combined vehicles move at 7.5 m/s after collision, and 187,500 J of kinetic energy is lost in the collision, converted primarily to heat, sound, and deformation of the vehicles.

Practice Problem 3: Classifying a Collision

Solution

Because the total kinetic energy of the system decreases from 12 J to 9 J, the collision is inelastic.

The missing 3 J of kinetic energy was transformed into other forms of energy such as thermal energy, sound, or deformation of the carts. This is characteristic of inelastic collisions — the system's total kinetic energy is not conserved, even though the total momentum of the system is still conserved.

If the total kinetic energy had remained at 12 J after the collision, it would have been classified as elastic. If the two carts had stuck together and moved with the same velocity after the collision, it would have been classified as perfectly inelastic.