Elastic Collisions

Introduction

In contrast to an inelastic collision, when kinetic energy is lost during the collision, an elastic collision occurs when numerous objects collide and the overall kinetic energy of the system is preserved. The law of momentum conservation applies to all sorts of collisions.

Most collisions in the real world result in the loss of kinetic energy in the form of heat and sound, therefore genuinely elastic physical impacts are uncommon. Some physical systems, on the other hand, lose very little kinetic energy and can thus be described as elastic collisions. Billiard balls colliding or the balls on Newton’s cradle are two of the most common examples of this.

Elastic Collision 

An elastic collision’s kinetic energy stays constant both before and after the contact. It is not converted into another kind of energy. It might be one-dimensional or two-dimensional. Perfectly elastic collisions are impossible in the actual world because there will always be some energy exchange, no matter how small. Even if the overall system’s linear momentum does not change, there is a change in the individual momenta of the relevant components, which are equal and opposite in magnitude and cancel each other out and the initial energy is preserved.

An elastic collision occurs when the system suffers no kinetic energy loss as a result of the collision. Momentum and kinetic energy are both preserved in an elastic collision.

A totally elastic collision happens when two bodies collide without losing any kinetic energy overall.

In the event of a collision, the kinetic energy is essentially unchanged before and after the impact and is not converted to any other sort of energy.

Examples: –

• A ball bounces back when we throw it on the floor. This is an example of an elastic collision in which momentum and kinetic energy are both preserved.
• The collision of the atoms is another example of an elastic collision.
• An elastic collision is shown by the collision of two billiard balls.

Characteristics

The following are the characteristics of an elastic collision:

• An object’s linear momentum is conserved in an elastic collision.
• The object’s overall energy is preserved.
• The kinetic energy of the system is also preserved.
• When an elastic collision occurs, conservative forces are at work.
• In an elastic collision, the mechanical energy is not transformed into heat.

Formula

Formula for elastic collision is,

M1U1 + M2U2  = M1V1 + M2V2  

Where-

First body mass = M1

Second body mass = M2

First body Initial Velocity = U1

Second body Initial Velocity = U2

First body Final Velocity = V1 

Second body Final Velocity = V2

Elastic collision kinetic energy is given by: 

½ M1U12 + ½ M2U22 = ½ M1V12+ ½ M2U22

Derivation of Formula for Elastic Collision

Elastic Collision momentum is calculated as follows:

M1U1 + M2U2  = M1V1 + M2V2  

∴ M(U1-V1) = M(V2-U2) —– (i)

The elastic collision kinetic energy formula is:

½ M1U12 + ½ M2U22 = ½ M1V12+ ½ M2U22

M2 = M1(U12-V12) (V22-U22)

M(U2-V2)(U2+V2)  = M(U1-V1)(U1+V1)  —— (ii)

Dividing the Eq (ii) to (i), we get

U1 + V1 = V2 + U2

U1 + U2 = -( V1 + V2)

Relative approach velocity = Relative receding velocity

Elastic Collision momentum is calculated as follows:

M1U1 + M2U2  = M1V1 + M2V2  

We already have,

U1 + V1 = V2 + U2

and, V2 = U1 + V1 – U2

In the momentum formula for Elastic Collision, enter the value of  V2

Now,

M1U1 + M2U2 = M1V1 + M2(U1 + V1 – U2)

or  M1U1 + M2U2 = M1V1 + M2 U1 + M2 V1 + M2 V1 – M2U2

or  M1U1 – M2 U1 +2M2U2 = V1 (M1 + M2)

Hence, V1 = ((M1 – M2) U1 + 2M2U2) / (M1+ M2)

Hence, V2 = ((M2 – M1) U2 + 2M1U1) / (M1+ M2)

Conclusion 

In contrast to an inelastic collision, when kinetic energy is lost during the collision, an elastic collision occurs when numerous objects collide and the overall kinetic energy of the system is preserved. An elastic collision occurs when the system suffers no kinetic energy loss as a result of the collision. In the event of a collision, the kinetic energy is essentially unchanged before and after the impact and is not converted to any other sort of energy. In an elastic collision, the mechanical energy is not transformed into heat.