Inelastic Collision is a type of collision where momentum is conserved, but kinetic energy is not. In such collisions, the colliding objects stick together, and some kinetic energy is transformed into other forms like vibrational energy or heat. This results in a loss of kinetic energy, which may transform into heat, sound, or deformation.

In this article, we will discuss all details related to inelastic collision such as definition, its types, examples, etc.

Inelastic Collision Definition

An inelastic collision is a type of collision in which momentum is conserved but kinetic energy is not. During an inelastic collision, the objects typically deform or stick together. This results in a loss of kinetic energy as some of it is transformed into heat, sound, or deformation.

In other words, the total kinetic energy of the system decreases after an inelastic collision. An example of an inelastic collision includes a collision between two cars that deforms upon impact.

Types of Inelastic Collision

There are 2 types of inelastic collision:

• Perfectly Inelastic Collision
• Partially Inelastic Collisions

Perfectly Inelastic Collision

A perfectly inelastic collision is a type of inelastic collision where two objects stick together after the collision and move as a single object. This means they have the same final velocity. Also in this collision loss of kinetic energy is maximum.

In such a collision, kinetic energy is lost by bonding the two bodies together, and this bonding energy usually results in a maximum kinetic energy loss of the system.

Note: In perfectly inelastic collision, Momentum is conserved but kinetic energy is not.

Partially Inelastic Collisions

Partially inelastic collisions refer to interactions between objects where the objects separate after the collision but are deformed due to the interaction. During such collisions, kinetic energy is not completely conserved, although momentum is still preserved.

The total initial kinetic energy of the objects involved in a partially inelastic collision is greater than the total final kinetic energy, meaning that some of the energy is transformed into other forms, such as thermal energy or sound energy. Examples include a ball dropping onto a hard surface and bouncing to a lower height compared to its original height before the collision.

Inelastic Collision Examples

The majority of collisions that occur in our daily lives are classified as inelastic collisions. Following is a list of a few of them.

• The ball is unable to rise to its initial height when it is dropped from a specific height.
• A ball of soft mud thrown against a wall will adhere to the wall.
• Two automobiles were involved in a collision.
• A vehicle colliding with a tree

Conservation of Momentum in Inelastic Collision

In an inelastic collision, momentum is conserved. This occurs as a result of the transfer of some kinetic energy to another object.

From the conservation of momentum, the formula during a collision is given by:

m₁v₁ + m₂v₂ = m₁v’₁ + m₂v’₂

If the collision is perfectly inelastic, the final velocity of the system is determined using

v’ = (m₁v₁ + m₂v₂)/m₁ + m₂

Kinetic Energy in Inelastic Collision

In an inelastic collision, kinetic energy is not conserved. Instead, some of the initial kinetic energy of the system is transformed into other forms of energy such as heat, sound, or deformation energy.

In a perfectly inelastic collision, where the colliding objects stick together and move as a single unit after the collision, the maximum amount of kinetic energy is lost.

Inelastic Collision Formula

We can only use momentum conservation as kinetic energy is not conserved. Since they stick together after collision, they move with one final velocity.

m₁v₁ + m₂v₂ = (m₁ + m₂)v

From this we can find the value of final velocity

v = (m₁v₁ + m₂v₂₎/m₁+m₂

For kinetic energy, [K.E. = 1/2 mv²]

1/2(m₁v₁² + m₂v₂²) > 1/2(m₁ + m₂)v²

Inelastic Collision in One Dimension

In a one-dimensional inelastic collision, some kinetic energy is lost, and the objects cling together afterward. “Perfectly inelastic” is another term used to describe this kind of collision.

A portion of the system’s kinetic energy is lost in a one-dimensional inelastic collision, and the colliding objects adhere to one another following the impact. The colliding items in a two-dimensional elastic collision move on a plane, and if the collision is elastic, the system’s kinetic energy is conserved.

Momentum is conserved in a one-dimensional inelastic collision, but internal kinetic energy is not. Almost all of the initial internal kinetic energy is lost in a collision that is perfectly inelastic. Most of this energy is transformed into thermal energy.

Inelastic Collision in Two Dimension

In a two-dimensional inelastic collision, objects have velocities in both the x and y directions, and momentum is conserved independently in each direction.

Conservation of momentum in the x-direction:

v_x = m₁v_1x + m₂v_2x / m₁+m₂

Conservation of momentum in the y-direction:

v_y = m₁v_1y+m₂v_2y/m₁+m₂

Since kinetic energy is not conserved in an inelastic collision, we typically need additional information to solve for the final velocities. This information could be in the form of coefficients of restitution.

Coefficient of Restitution

It is defined as the ratio of relative velocity of separation to relative velocity of approach.

Let two particles m₁ , m₂ be moving with initial velocities u₁ , u₂ and after collision moving with velocities v₁ , v₂

e = | v₂ – v₁ |/| u₂– u₁ |

It is also defined as the square root of the ratio of final kinetic energy to initial kinetic energy.

e = √(Final KE/Initial KE)

Different values of e for different types of collisions:

• e = 0 : Perfectly inelastic collision
• 0 < e < 1 : Inelastic collision
• e = 1 : Elastic collision

Read more about Elastic Collision Formula.

Elastic Collision vs Inelastic Collision

Lets discuss the difference between Elastic collision vs Inelastic Collision