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# Inelastic Collision

Collision is a free event which takes place between the two or more than molecules. This is completed within a short time period. The concept of collision is based on the conservation of momentum and change in the kinetic energy. These laws imply that there is no change in the total momentum and kinetic energy of the system when the collision takes places. It is classified in two types,
• Elastic collision
• Inelastic collision.
The elastic collision involves conservation of momentum and kinetic energy without dissipative force. The collision between hard steel balls or swinging balls, ideal gases collision and also the collision of atomic or nuclear scattering is elastic in nature. The inelastic collision is the collision which involves only change in kinetic energy. All ordinary collisions are inelastic in nature because in this collision, the kinetic energy is changed into internal energy; for examples, collision between car, trucks or airlines etc.

Here, we are discussing about inelastic collision in one dimension and two dimensions, a perfect collision, and its mathematical formulation.

## Inelastic Collision Definition

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A collision is said to be an inelastic collision, if the kinetic energy is not conserved in the collision. However, the momentum is conserved.
• In these collisions the kinetic energy of the system of objects is not conserved after the collision.
• The resultant momentum of the bodies that are involved in the collision remains the same.
• In these types of collisions some of the initial kinetic energy is converted into the heat energy.
• This heat energy is utilized in a little deformation of the bodies or in some other work or utilized in other manner.
• The collisions which tend to slow down the involved bodies or leads to them sticking together are called inelastic collisions.
• In these collisions some kinetic energy is utilized in the vibration of the atoms of the objects involved.

The kinetic energy lost in the collision appears in the form of heat energy, sound energy or light energy. The forces of interaction in an inelastic collision are non-conservative in nature.

Most of the collisions between macroscopic bodies are inelastic collisions. If a ball is dropped from a certain height and it is unable to rise to its original height, it would mean that ball has lost some kinetic energy (which would appear as heat energy). This would mean that collision is an Inelastic Collision.

## Perfectly Inelastic Collision

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If the two bodies stick together after the collision, the collision is said to be perfectly inelastic. In this type of collision, the loss of kinetic energy is maximum but not complete. In a perfectly inelastic collision there are certain points to be kept in mind:
1. The coefficient of restitution is zero.
2. The colliding objects stick together after the collision.
3. The momentum is conserved.
4. Kinetic energy is not conserved.

## Inelastic Collision Formula

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The equation that denotes the conservation of momentum is:

$m_{1}u_{1}+m_{2}u_{2}$ = $(m_{1}+m_{2})v$

Here, m1 = mass of object or body 1
m2 = mass of object or body 2
u1 = initial velocity of object or body 1
u2 = initial velocity of object or body 2
v = final velocity of both the objects

The final velocity is given by v = $\frac {m_{1}u_{1} + m_{2} u_{2}} {(m_{1}+m_{2})}$

The loss in kinetic energy $E$ = $\frac{1}{2}$ $m_{1} u_{2}^{2}$ - $\frac{1}{2}$ $(m_{1} + m_{2}) v^{2}$

or $E$ = $\frac{1}{2}$ $(m_{1}u_{1}^{2} + m_{2}u_{2}^{2})$ - $\left ( \frac {m_{1}u_{1} + m_{2}u_{2}} {m_{1} + m_{2}} \right ) ^{2}$

The velocities after a one dimensional Inelastic Collision Equation are given by:

$Va$ = $\frac {[Cr Mb ( Ub – Ua) + Ma Ua + Mb Ub]} {Ma + Mb}$ . . . . . . . . . . (1)

$Vb$ = $\frac {[Cr Mb ( Ua – Ub) + Ma Ua + Mb Ub]} {Ma + Mb}$ . . . . . . . . . . . (2)

Here: Cr = Coefficient of restitution
Ma = Mass of object a
Mb = Mass of object b
Ua = The initial velocity of object a or velocity before impact
Ub = The initial velocity of object b.

Also Cr = 0 for a perfectly inelastic collision and 1 for elastic collisions.

If we consider the center of momentum frame than the formula reduces to:

Va = -Cr Ua = = = = = = = = => (3)

Vb = -Cr Ub = = = = = = = = => (4)

## Inelastic Collision in One Dimension

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For handling inelastic collision in one dimension we need to invoke the law of conservation of momentum. It is presumed that there is no external force acting on the system of objects or bodies.

The velocities of objects a and b after an one dimensional inelastic collision are given by:

$Va$ = $\frac {[Cr Mb ( Ub – Ua) + Ma Ua + Mb Ub]} {Ma + Mb}$

$Vb$ =
$\frac {[Cr Mb ( Ua – Ub) + Ma Ua + Mb Ub]} {Ma + Mb}$

For conserved momentum:

We have:
$P_{before}$ = $P_{after}$

$P_{before}$ = $M_{1}V_{1}$ + $M_{2}V_{2}$

$P_{after}$ = $M_{1}V$ + $M_{2}V$

Hence: $M_{1}V_{1}$+ $M_{2}V_{2}$ = $(M_{1}$ + $M_{2})V$

or V = $\frac {(M_{1} V_{1} + M_{2} V_{2})} {(M_{1} + M_{2})}$

There is no conservation of kinetic energy as such.

(i) The max transfer of energy occurs when m1 = m2,
(ii) If Ki and Kf are the initial and final kinetic energies of mass m1, the fractional decrease in its kinetic energy is given by
= $\frac {k_{i}} {k_{f}}$ = 1- $\frac {v_{1}^{2}} {u_{1}^{2}}$

Further, if m2 = nm1 and u2 = 0, then

= $\frac {k_{i}-k_{f}} {k_{i}}$ = $\frac {4n} {(1+n)^{2}}$

## Inelastic Collision Examples

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Various examples of inelastic collision are:

1. Car crash : the car crashing against the tree is the example of inelastic collision as the kinetic energy is not conserved here. The car stops. 2. The collision between two objects such that after the collision both stick to each other and hence move with the same velocity. 3. The bullet hitting wood is also an example of inelastic collision. The kinetic energy is not conserved in this case also. 4. Seat belt tied to a person and the sudden application of breaks is also an example of inelastic collision.

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