Elastic Collision Formula

Elastic Collision is the collision in which the kinetic energy of the system is conserved. A collision occurs when two or more object comes in direct contact with each other. the collision can be of two types that includes, Elastic Collision and Inelastic Collision.

In this article, we will learn about, Elastic Collision, Examples of Elastic Collision, Difference between Elastic Collision and Inelastic Collision, and others in detail.

What is Elastic Collision?

A fully elastic collision occurs when two bodies come into contact without losing any overall kinetic energy. An elastic collision is one in which the system loses no kinetic energy due to the collision. In an elastic collision, momentum and kinetic energy are both conserved.

In the case of a collision, the kinetic energy before and after the collision is essentially unchanged and is not transferred to any other type of energy.

The kinetic energy of an elastic collision remains constant before and after the contact. It isn’t changed into another type of energy. It might be one-dimensional or two-dimensional in nature. In the real world, perfectly elastic collisions are impossible because there will always be some energy exchange, no matter how minor. There is a change in the individual momenta of the relevant components, which are equal and opposite in size and cancel each other out, and the original energy is conserved, even though the linear momentum of the overall system does not change.

Elastic Collision Definition

Eleastic collision is defined as the collision in which the kinetic energy of the system is conserved. In elastic collision momentum of the system is also conserved.

Inelastic Collision Definition

Inelastic collision is the collision in which the kintenic energy of the system is not conserved. For example, the kicking of ball is an example of Inelastic collision.

Examples of Elastic Collision

Various examples of the elastic collision are,

• A ball bounces back when it is thrown on the floor. It’s because a moving ball maintains its overall momentum and kinetic energy.
• When two atoms collide, they produce an elastic collision, however, it is only termed an elastic collision if no energy is lost.
• The striking of balls with the stick while playing pool or snooker is a simple illustration of elastic collision.

Elastic Collision Formula

The momentum formula for Elastic Collision is,

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

where,

• m₁ is the Mass of 1^st body
• m₂ is the Mass of 2^nd body
• u₁ is the Initial Velocity of 1^st body
• u₂ is the Initial Velocity of 2^nd body
• v₁ is the Final Velocity of 1^st body
• v₂ is the Final Velocity of 2^nd body

The kinetic energy formula for elastic collisions is,

1/2(m₁u₁²) + 1/2(m₂u₂²) = 1/2(m₁v₁²) + 1/2(m₂v₂²)

Elastic Collision Formula Derivation

Momentum formula for Elastic Collision is:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

∴ m(u₁-v₁) = m(v₂-u₂)  ⇢ (Equation A)

Kinetic energy formula for elastic collisions is:

1/2(m₁u₁²) + 1/2(m₂u₂²) = 1/2(m₁v₁²) + 1/2(m₂v₂²)

∴ m₁(u₁²-v₁²) = m₂(v₂²-u₂²)

∴ m₁(u₁+v₁)(u₁-v₁) = m₂(v₂+u₂)(v₂-u₂)  ⇢ (Equation B)

Divide Equation B to Equation A,

u₁ + v₁ = v₂ + u₂

∴ u₁ – u₂ = -(v₁ – v₂)

Relative velocity of Approach = Relative Velocity of Receding

Momentum formula for Elastic Collision is:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

We have,

u₁ + v₁ = v₂ + u₂

∴ v₂ = u₁ + v₁ – u₂

Put v₂ in the momentum formula for Elastic Collision,

∴ m₁u₁ + m₂u₂ = m₁v₁ + m₂(u₁ + v₁ – u₂)

∴ m₁u₁ + m₂u₂ = m₁v₁ + m₂u₁ + m₂v₁ – m₂u₂

∴ m₁u₁ – m₂u₁ +2m₂u₂ = v₁(m₁ + m₂)

v₁ = ((m₁ – m₂)u₁ + 2m₂u₂) / (m₁ + m₂)

v₂ = ((m₂ – m₁)u₂ + 2m₁u₁) / (m₁ + m₂)

Elastic Vs Inelastic Collision

The basic difference between and Elastic Collision and Inelastic Collision are shown in the table added below,

Elastic Collision	Inelastic Collision
Total kinetic energy is conserved in Elastic Collision	Total kinetic energy is not conserved in Inelastic Collision
Momemtum is conserved in case of Elastic Collision	Momemtum is not conserved in case of Inelastic Collision
In does not occur in general cases and is rarely seen in our daily life.	It occurs in our daily life and is seen in various incident in our daily life.
Examples of Elastic Colliion are, collision of atoms in the air, etc.	Examples of Inelastic collision are collision of two cars, kicking of ball, etc.

The basic differences between Elastic and Inelastic Collision is shown in the image added below,

Applications of Elastic Collision

The following are some of the applications for elastic collisions:

• The amount of force experienced by the body during the collision is affected by the amount of time involved in the collision. In other words, the smaller the force acting on the body, the longer the impact takes. As a result, in order to optimize the force, the collision duration must be reduced.
• However, in order to reduce the force, the contact duration must be increased. There are numerous examples of this. One of them is that airbags in cars reduce the amount of time spent in a collision and reduce the force applied to the item. The phenomenon behind this procedure is that the airbags reduce the force while increasing the object’s crash time.
• Although an elastic collision can occur in a variety of locations or settings, inelastic collisions are more common.

Read More,

• Law of Conservation of Mometum
• Potential Energy
• Velocity

Elastic Collision Formula Examples

Example 1: A 5 kg ball moving east at a speed of 6 m/s strikes a 2 kg ball at rest. Calculate the velocities of the two balls assuming a perfectly elastic collision.

Solution:

Given,

• m₁ = 5 kg,
• u₁ = 6 m/s,
• m₂ = 2 kg,
• u₂ = 0

To Find, v₁, v₂

We have,

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

∴ (5 × 6) + (2 × 0) = 5v₁ + 2v₂

∴ 30 + 0 = 5v₁ + 2v₂

∴ 30 = 5v₁ + 2v₂  ⇢ (Equation 1)

u₁ + v₁ = u₂ + v₂

∴ 6 + v₁ = 0 + v₂

∴ 6 = -v₁ + v₂  ⇢ (Equation 2)

From Equation 1 and 2,

∴ v₂ = 8.57 m/s

Put v₂ = 8.57 in Equation 2,

-v₁ + 8.57 = 6

∴ v₁ = 2.57 m/s

Example 2: When a body of mass 4 kg collides with another body at rest, it continues to move in the same direction at a speed of one-third of its original speed. Determine the second body’s mass.

Solution:

Given,

• m₁ = 4 kg
• v₁ = u₁/3
• u₂ = 0

To Find, m₂

We have,

v₁ = ((m₁ – m₂) / (m₁ + m₂)) × u₁ + (2m₂u₂) / (m₁ + m₂)

∴ u₁/3 = ((4 – m₂) / (4 + m₂)) × u₁

∴ 4 + m₂ = 12 – 3m₂

∴ 4m₂ = 12 – 4

∴ m₂ = 2 kg

Example 3: A 1 kg ball travelling at 15 m/s collides head-on with another ball of unknown mass at rest, rebounding with a velocity of 10 m/s. What is the other ball’s mass?

Solution:

Given,

• m₁ = 1 kg
• u₁ = 15 m/s
• v₁ = -10 m/s (rebounds)
• u₂ = 0

to Find, m₂

We have,

v₁ = ((m₁ – m₂) / (m₁ + m₂)) × u₁ + (2m₂u₂) / (m₁ + m₂)

∴ -10 = ((1 – m₂) / (1 + m₂)) × 15

∴ -2 + (-2m₂) = 3 – 3m₂

∴ m₂ = 5 kg

Example 4: Assume that m₁ is 3 kg, m₂ is 5 kg, u₂ is at rest, v₁ is 2.2 m/s, and v₂ is 2 m/s in a specific example. What is the value of u₁?

Solution:

Given,

• m₁ = 3 kg
• m₂ = 5 kg
• u₂ = 0
• v₁ = 2.2 m/s
• v₂ = 2 m/s

To Find, u₁

We have,

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

∴ 3 × u₁ + 5 × 0 = 3 × 2.2 + 5 × 2

∴ 3u₁ = 6.6 + 10

∴ u₁ = 5.53 m/s

Example 5: If u₁ is 4 m/s, u₂ is 2 m/s, and v₂ is 6 m/s, the following equation can be used. What is the significance of v₁?

Solution:

Given,

• u₁ = 4 m/s
• u₂ = 2 m/s
• v₂ = 6 m/s

To Find, v₁

u₁ + v₁ = v₂ + u₂

∴ 4 + v₁ = 6 + 2

∴ v₁ = 8 – 4

∴ v₁ = 4 m/s

Practice Questions on Elastic Collisions

Q1. If u₁ is 12 m/s, u₂ is 5 m/s, and v₂ is 8 m/s, then find the v₁ of the collision if the mass of object is m.

Q2. If u₁ is 8 m/s, u₂ is 10 m/s, and v₂ is 6 m/s, the following equation can be used. What is the significance of v₁?

Q3. A box of mass 12 kg is sliding at 15 m/s collides head-on with another box of unknown mass at rest, rebounding with a velocity of 10 m/s. What is the mass of other box?

Q4. When a body of mass 12 kg collides with another body at rest, it continues to move in the same direction at a speed of one-fourth of its original speed. Determine the mass of second body.

Elastic Collision-FAQs

1. What is Elastic Collision?

In the collision if there is no net loss in Kinteic Energy in the system then the collision is called the Elastic Collision.

2. What is Elastic Collision Example?

The collision of gas particle in the atmosphere is an example of Elastic Collsion because there is no net loss of energy in the collsion.

3. What is Difference between Elastic and Inelastic Collision?

In Elastic collision the net kinetic energy of the system is conserved, whereas in case of inelastic collision the net kinetic energy of the system is not conserved.

4. What is Inelastic Collision?

In the collision if there is a net loss in Kinteic Energy in the system then the collision is called the Inelastic Collision.

5. What is Inelastic Collision Example?

Various examples of the Inelastic Collision are,

• Collision of two cars.
• Bouncing of ball from the ground.
• Kicking of ball, etc.