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Inelastic Collision

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More often you see the objects undergo collision like a billiard balls collides, a man accidentally dashes another man.
These collisions are basically categorized into two types:
1. Elastic Collision
2. Inelastic Collision

Sometimes you could notice bodies sticking together after collision like a balls collides a wall and breaks, a car getting smashed by a heavy loaded truck etc. This is what inelastic collision is! Here energy gets transferred from a body to the other however the momentum gets conserved.

What is Inelastic Collision?

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In an Inelastic collision the kinetic energy is not conserved. The total momentum of two bodies remains the same but the initial kinetic energy is converted into heat. Here the particles stick together on impact and hence the loss of kinetic energy takes place. Common velocity of the body after collision is given by
m1 v1 + m2 v2 = (m1 + m2)vf
or
vf = $\frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$
When second body is at rest v2 = 0 then
vf = $\frac{m_1}{m_1 + m_2}$ v1

Perfectly Inelastic Collision

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Consider two bodies of masses m1 and m2 with initial velocity v1 and v2 along a straight line. After collision if these two bodies stick on together the common velocity is v then the collision is perfectly inelastic. The momentum before collision is equal to the momentum after collision.

By law of conservation of momentum
m1 v1 + m2 v2 = (m1 + m2)v2
Since mass m2 is at rest v2 = 0. Hence
m1v1 = (m1 + m2) vThe final speed is
v = $\frac{m_1 v_1}{m_1 + m_2}$

Momentum Inelastic Collision 

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If no external forces acts on a system momentum is conserved in an Inelastic collision
The total momentum before collision = m1v1 + m2v2
The total momentum after collision = (m1 + m2)v

If the momentum is conserved then
pbefore Collision = pafter Collisionso
m1v1 = (m1 + m2)v

Hence v = $\frac{m_1 v_1}{m_1 + m_2}$
This is Momentum Inelastic Collision formula

Inelastic Collision Formula

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Let us take a body of mass m1 moving with velocity v1 and mass m2 moving with velocity v2 undergoes the inelastic Collision as shown in fig. Then their kinetic energy is given by energy before Collision = $\frac{1}{2}$ m1 v12 and energy after Collision = $\frac{1}{2}$ (m1 + m2) v2.

The loss of kinetic energy is the kinetic energy after collision minus the kinetic energy before collision i.e.,

K.Ei - K.Ef = $\frac{1}{2}$ m1v12 - $\frac{1}{2}$ (m1 + m2) v2

= $\frac{1}{2}$ m1v12 - $\frac{1}{2}$ (m1 + m2) $(\frac{m_1 v_1}{m_1 + m_2})^2$

= $\frac{1}{2}$ m1v12 - $\frac{1}{2}$ $\frac{m_1^2 v_1^2}{m_1 + m_2}$

= $\frac{1}{2}$ $\frac{m_1 v_1^2 (m_1 + m_2) - m_1^2 v_1^2 }{m_1 + m_2}$

= $\frac{1}{2}$ $\frac{m_1^2 v_1^2 + m_1 m_2 v_1^2 - m_1^2 v_1^2}{m_1 + m_2}$

= $\frac{m_1 m_2}{m_1 + m_2}$ v12.

A One Dimensional Inelastic Collision

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The velocities after a one dimensional Collision equation is given by
va = $\frac{c_r M_b (U_b - U_a) + M_a U_a + M_b U_b}{M_a + M_b}$

vb = $\frac{c_r M_b (U_a - U_b) + M_a U_a + M_b U_b}{M_a + M_b}$
Where Cr = Coefficient of restitution
Ma = Mass of body a
Mb = Mass of body b
Ua = Initial Velocity of body a
Ub = Initial velocity of body b.

2D Inelastic Collision

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Lets us consider the inelastic Collision in 2 dimension. If vi is the initial velocity of mass m1 colliding with mass m2 having velocity v2. It will be having two dimension x and y given as:

The Inelastic Collision in x-direction is
m1 v1ix + m2 v2ix = (m1 + m2) vfx
The Inelastic Collision in y-direction is
m1 v2iy + m2 v2iy = (m1 + m2) vfy

Inelastic Collisions Examples

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Here are some examples of Inelastic Collisions
  1. Crash taking place between the two vehicles
  2. A Bullet hitting the target
  3. A man in hurry clashes with another guy coming in front of him
  4. A boy hitting the ball