Elastic And Inelastic Collision In One And Two Dimension

In an elastic collision, both kinetic energy and momentum are conserved. The total energy before and after the collision remains constant while in inelastic collisions  momentum is conserved, but kinetic energy is not. Some kinetic energy is converted into other forms of energy such as heat or sound. Understanding these collision types helps in analyzing various physical situations and solving problems in physics.

1.0Definition of Collision

• When two particles approach each other and share common space then their momentum gets redistributed and this phenomenon is called collision.
• Particles approach each other just before collision, means they have velocity of approach
• Particles get separated after collision means they have velocity of separation. 

2.0Line of motion And Line of Impact

1. Line of motion(LOM): The line along which particles are moving just before collision.

Ex.- 1 Objects moving along same line

Ex. - 2 Objects moving in same plane but along different line

2. Line of impact(LOI): Line along which Transfer  of momentum takes place  it is line along common normal as normal forces are responsible for redistribution of momentum

3.0Types of Collision

1. Head-on collision or 1-D collision

• If line of motion and line of impact are same then it is head on collision
• In head on collision particles move along same line before and after collision
• In head on collision initial velocities of colliding objects lie along common normal

2. Oblique collision or 2-D collision

• If line of motion and line of impact are not same then it is Oblique  collision
• In Oblique  collision particles move along different line before and after collision

4.0Process of Collision and Coefficient of Restitution

• To understand the mechanism of a collision, let us consider two balls A and B of masses m_A and m_B moving with velocities u_A and u_B in the same direction as shown. Velocity u_A is larger than u_B so that ball A hits the ball B.
• During the impact, both the bodies push each other and first they get deformed till the deformation reaches a maximum value and then they try to regain their original shapes due to elastic behaviour of the materials the balls formed of.

No external forces $\Rightarrow$ Principle of Conservation of Momentum is applicable

• During collision

5.0Deformation period

• The time interval during which deformation takes place.
• During the period of deformation, due to push applied by the balls on each other, speed of ball A decreases and that of ball B increases.
• At the end of the deformation period, when the deformation is maximum both the balls move with the same velocity say it is u. Thereafter, the balls will either move together with this velocity or follow the period of restitution.

6.0Maximum Deformation

• at this instant balls are moving with same velocity 

$v_1=v_2=v_{com}$

• at maximum Deformation KE of system is minimum

$(KE{)}_{min}=\frac{1}{2}{m}_{system}{v}_{com}^2$

max loss in KE possible = $\frac{1}{2}\mu u_{relative}^2$

7.0Reformation or Restitution Period

• In this period balls are recovering from deformation
• The time interval in which the bodies try to regain their original shapes.
• During the period of restitution due to push applied by the balls on each other, speed of ball A decreases further and that of ball B increases further till they separate from each other.

8.0Coefficient of Restitution:

• Usually the force D applied by the bodies A and B on each other during the period of deformation differs from the force R applied by the bodies on each other during the period of restitution. 
• Therefore, it is not necessary that the magnitude of impulse $\int Ddt$ deformation equals that of impulse $\int Rdt$ due to restitution
• The ratio of magnitudes of impulse of restitution to that of deformation is called the coefficient of restitution and is denoted by e. 
• $e=\frac{impulseofrecovery}{impulseofdeformation}=\frac{\int Rdt}{\int Ddt}$
• $e=\frac{velocityofseparationalonglineofimpact}{velocityofapproachalonglineofimpact}=\frac{|v_B-v_A|}{|u_A-u_B|}$

Note: Coefficient of restitution depends on various factors such as elastic properties of materials forming the bodies. In general, Based upon the extent of Reformation { which is due to elastic forces) collision is characterized in three categories.

9.0Perfectly Elastic Collision

• For these collision $e=\frac{v_2-v_1}{u_1-u_2}=1$
• reformation =deformation 
• All  of the maximum PE of deformation is converted back to KE and  Energy loss is zero and thus initial KE of the system before and after collision is the same.

initial KE of system= final KE of system

• (note that it is same not conserved as during collision it gets converted to PE of deformation and again converted to KE thus not conserved at each instant before and after collision)
• Velocity of separation=velocity of approach
• The Momentum is conserved.
• Total energy is conserved.

10.0Perfectly Plastic or Inelastic Impact

• For these impacts e = 0
• Reformation =0
•  Bodies undergoing impact move with same velocity after the impact. V₁ = V₂ = V_com
• Energy loss = $\frac{1}{2}\mu u_{rel}^2$ where $\mu$ is reduced mass $\mu =\frac{m_1{m}_2}{m_1+m_2}$ and $u_{rel}$ is initial relative velocity.

11.0Inelastic Impact

• For these impacts 0<e<1 
• reformation <deformation
• Energy lost is less than maximum possible loss.

12.0Elastic Collision in Two Dimensions(Oblique Collision)

A collision in which the particles move in the same plane at different angles before and after the collision is called Oblique Collision.

By COLM along x-axis,

$m_1{u}_1+m_2{u}_2=m_1{v}_{1}Cos\theta +m_2{v}_{2}Cos\varphi$

By COLM along y-axis

$0+0=m_1{v}_{1}Sin\theta -m_2{v}_{2}Sin\varphi$

If Collision is Elastic then, by Conservation of Kinetic Energy

$\frac{1}{2}{m}_1{u}_1^2+\frac{1}{2}{m}_2{u}_2^2=\frac{1}{2}{m}_1{v}_1^2+\frac{1}{2}{m}_2{v}_2^2$

Oblique Collision:-(Smooth Surface)

• (LOI≠LOM)
• This redistribution of momentum takes place along common normal(LOI).
• Momentum perpendicular to LOI is Constant.

$v_{1}Sin\theta =v_{2}Sin\alpha$………….(1)

$e=\frac{v_{2}Cos\alpha }{v_{1}Cos\theta }$

$e{v}_{1}cos\theta =v_{2}cos\alpha$………..(2)

Dividing equation (1) by (2)

$\frac{Tan\theta }{e}=Tan\alpha$

$Tan\theta =eTan\alpha$

• when two bodies of same mass collide elasticity & obliquely then they are more perpendicular to each other after collision.

13.0Difference Between Elastic and Inelastic Collision 

14.0Sample Questions on Elastic and Inelastic Collision

Q.1 Show that linear momentum is conserved during a collision?

Sol. Suppose two bodies 1 and 2 collide against each other. They exert mutual impulsive forces on each other during the collision time $\Delta$t. The changes produced in the momenta of the two bodies will be,

$\Delta \overrightarrow{p_1}=\overrightarrow{F_{12}}\Delta t$ and $\Delta \overrightarrow{p_2}=\overrightarrow{F_{21}}\Delta t$

According to Newton third Law,

$\overrightarrow{F_{12}}=-\overrightarrow{F_{21}}$

$\overrightarrow{F_{12}\Delta t}=-\overrightarrow{F_{21}}\Delta t$

$\overrightarrow{F_{12}}\Delta t+\overrightarrow{F_{21}}\Delta t=0$

$\Delta \overrightarrow{p_1}+\Delta \overrightarrow{p_2}=0$

$\Delta (\overrightarrow{p_1}+\overrightarrow{p_2})=0$

$\overrightarrow{p_1}+\overrightarrow{p_2}=\text{Constant}$

Q.2 Find the value of e for the following collision?

Solution:

$\frac{0.5}{1}=0.5$

Q.3 Find min KE & max PE stare during the following elastic collision?

Solution:

$K.E_{min}=\frac{1}{2}(3)(\frac{4}{3}{)}^2=\frac{8}{3}Joule$

$P.E_{max}=\frac{1}{2}(\frac{2}{3})(1{)}^2=\frac{1}{3}Joule$