Inelastic Collision

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

Criteria	Elastic Collision	Inelastic Collision
Kinetic Energy	In elastic collisions, the kinetic energy is the same before and after the collision.	In inelastic collisions, the kinetic energy is not conserved.
Deformation	In elastic collisions, the masses involved don’t deform or stick together.	In inelastic collisions, they get deformed, and in perfectly inelastic collisions they stick together and move forward as one.

Conclusion: Inelastic Collision

In collisions, two types exist: elastic and inelastic. Elastic collisions conserve both momentum and kinetic energy, while inelastic collisions conserve momentum only. Perfectly inelastic collisions are a special case where objects stick together after collision. Momentum is conserved in both one-dimensional and two-dimensional inelastic collisions. The coefficient of restitution quantifies energy retention after collision.

Solved Examples

Example 1: Two objects, A and B, with masses m₁= 2 kg and m₂= 3 kg respectively, collide inelastically. Object A is initially moving to the right with a velocity of v₁ = 4m/s, while object B is initially at rest. After the collision, they move together as one mass. Find the final velocity and the coefficient of restitution for this collision.

Solution:

We simply use the formula :

v=m₁v₁+m₂v₂/m₁+m₂ = 2.4+3.0/ 2+3= 1.6 m/s

Since they stick together after collision, this is a perfectly inelastic collision.

Therefore, the coefficient of restitution for this collision is e = 0.

Example 2: Two objects, A and B, with masses m₁= 0.2 kg and m₂= 0.3 kg respectively, collide inelastically. Object A is initially moving to the right with a velocity of u₁ = 4m/s, while object B is initially at rest. After collision object A rebounds with a velocity of v₁= 2m/s. Find the coefficient of restitution for this collision.

Solution:

We can use the conservation of momentum to find the velocity of B after collision :

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

0.2 × 4 + 0 = 0.2 × 2 + 0.3 × v₂

This gives the velocity of B

v₂ = 4/3 = 1.33 m/s

The coefficient of restitution is given by the formula

e = |v₂ – v₁ |/ |u₂– u₁ | = [(4/3) -2 ]/[0-4] = 1/6

Therefore, the coefficient of restitution for this collision is e = 1/6

Practice Problems on Inelastic Collision

Problem 1: A 0.2 kg bullet moving with a velocity of 500m/s collides inelastically with a 0.5 kg wooden block initially at rest on a frictionless surface. After the collision, the bullet and the block move together. Calculate their final velocity.

Problem 2: Two railway cars, one with a mass of 1500 kg and the other with a mass of 2500 kg, collide inelastically. The lighter car is moving to the right with a velocity of 5m/s, while the heavier car is moving to the left with a velocity of 3 m/s. After the collision, the lighter car moves left with a final velocity of 1.2 m/s. Determine the final velocity of the heavier car and the coefficient of restitution.

Problem 3: A tennis ball of mass 0.06 kg is dropped from a height of 2 m onto a concrete floor. After the bounce, the ball rebounds to a height of 1.5 m. Calculate the coefficient of restitution for this collision.

FAQs on Inelastic Collision

What is Inelastic Collision?

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.

What is a perfectly Inelastic Collision?

Perfectly inelastic collision is a specific type of inelastic collision where the objects stick together after the collision.

Differentiate between Elastic and Inelastic collisions.

The main differences between elastic and inelastic collisions lie in the conservation of kinetic energy and deformation or sticking together of objects. Kinetic energy is conserved for elastic collisions only, and sticking or deformation occurs in inelastic cases only.

Is Momentum Conserved in Inelastic Collision?

Yes, momentum is conserved in an inelastic collision.

Is Kinetic Energy Conserved in Inelastic Collision?

In an inelastic collision, kinetic energy is not conserved. Some kinetic energy is lost and transferred to another object in an inelastic collision. Friction within the system is the cause of this kinetic energy loss.

What are Some Example of Perfectly Inelastic Collision?

Here are a few instances of collisions that are perfectly inelastic: Following an accident, two railway cars stay together. When two balls of clay collide and stick together.