Inelastic Collision

In physics, collisions are considered inelastic when the colliding objects do not conserve kinetic energy, although they do conserve momentum. During an inelastic collision, some of the kinetic energy of the colliding bodies is transformed into other forms of energy, such as heat, sound, or deformation energy.

Conservation of Momentum

In any collision, the total momentum of the system is conserved, provided no external forces are acting on the system. This is expressed by the conservation of momentum law:

$$ \vec{p}{\text{initial}} = \vec{p}{\text{final}} $$

Where $\vec{p}$ represents momentum, which is the product of mass ($m$) and velocity ($\vec{v}$):

$$ \vec{p} = m \vec{v} $$

For a system of two objects colliding inelastically, the conservation of momentum can be written as:

$$ m_1 \vec{v}{1i} + m_2 \vec{v}{2i} = (m_1 + m_2) \vec{v}_f $$

Here, $m_1$ and $m_2$ are the masses of the two objects, $\vec{v}{1i}$ and $\vec{v}{2i}$ are their initial velocities, and $\vec{v}_f$ is their final velocity after the collision.

Energy Consideration

In an inelastic collision, the kinetic energy is not conserved. The initial kinetic energy is greater than the final kinetic energy:

$$ KE_{\text{initial}} > KE_{\text{final}} $$

The kinetic energy ($KE$) of an object is given by:

$$ KE = \frac{1}{2} m v^2 $$

The loss of kinetic energy in an inelastic collision goes into other forms of energy, as mentioned earlier.

Types of Inelastic Collisions

There are two main types of inelastic collisions:

1. Partially Inelastic Collision: Some kinetic energy is lost, but the objects do not stick together.
2. Perfectly Inelastic Collision: The maximum amount of kinetic energy is lost, and the objects stick together and move with a common velocity after the collision.

Differences and Important Points

Here is a table summarizing the key differences between elastic and inelastic collisions:

Aspect	Elastic Collision	Inelastic Collision
Kinetic Energy	Conserved	Not conserved
Momentum	Conserved	Conserved
Final State	Objects rebound off each other	Objects may stick together or separate
Energy Transformation	None or negligible	Transformed into heat, sound, deformation
Examples	Billiard balls hitting each other	Car crash where vehicles crumple

Formulas

For a perfectly inelastic collision, where two objects stick together, the final velocity can be found using the conservation of momentum:

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

Examples

Example 1: Perfectly Inelastic Collision

Imagine a 5 kg cart moving at 2 m/s collides with a stationary 10 kg cart, and they stick together. The final velocity of the combined carts can be calculated as follows:

$$ \vec{v}_f = \frac{(5 \text{ kg})(2 \text{ m/s}) + (10 \text{ kg})(0 \text{ m/s})}{5 \text{ kg} + 10 \text{ kg}} = \frac{10 \text{ kg} \cdot \text{m/s}}{15 \text{ kg}} = \frac{2}{3} \text{ m/s} $$

Example 2: Partially Inelastic Collision

Suppose a 3 kg ball moving at 4 m/s hits a 2 kg ball at rest, and after the collision, the 3 kg ball moves at 1 m/s in the same direction. To find the velocity of the 2 kg ball after the collision, we use the conservation of momentum:

$$ (3 \text{ kg})(4 \text{ m/s}) + (2 \text{ kg})(0 \text{ m/s}) = (3 \text{ kg})(1 \text{ m/s}) + (2 \text{ kg})\vec{v}_{2f} $$

Solving for $\vec{v}_{2f}$:

$$ 12 \text{ kg} \cdot \text{m/s} = 3 \text{ kg} \cdot \text{m/s} + 2 \text{ kg} \cdot \vec{v}_{2f} $$

$$ \vec{v}_{2f} = \frac{9 \text{ kg} \cdot \text{m/s}}{2 \text{ kg}} = 4.5 \text{ m/s} $$

The 2 kg ball moves away at 4.5 m/s after the collision.

In conclusion, understanding inelastic collisions is crucial for analyzing situations where kinetic energy is not conserved, but momentum is. This concept is widely applicable in various fields, including automotive safety, sports, and material science.