ELASTIC AND INELASTIC COLLISION


ELASTIC COLLISION



An elastic collision is that in which the momentum of the system as well as kinetic energy of the system before and after collision is conserved.  
INELASTIC COLLISION


An inelastic collision is that in which the momentum of the system before and after collision is conserved but the kinetic energy before and after collision is not conserved.  
ELASTIC COLLISION
IN ONE DIMENSION 


Consider two nonrotating spheres of mass m_{1} and m_{2} moving initially along the line joining their centers with velocities u_{1 }and u_{2 }in the same direction. Let u_{1 }is greater than u_{2}. They collide with one another and after having an elastic collision start moving with velocities v_{1} and v_{2} in the same directions on the same line.  


Momentum of the system after collision = m_{1}v_{1} + m_{2}v_{2}


According to the law of conservation of momentum:  
m_{1}u_{1} + m_{2}u_{2 }= m_{1}v_{1} + m_{2}v_{2}
m_{1}v_{1} – m_{1}u_{1 }= m_{2}u_{2} – m_{2}v_{2} m_{1}(v_{1} – u_{1}) = m_{2}(u_{2} – v_{2}) ——(1) 

Similarly  
K.E of the system before collision = ½ m_{1}u_{1}^{2} + ½ m_{2}u_{2}^{2}
K.E of the system after collision = ½ m_{1}v_{1}^{2} + ½ m_{2}v_{2}^{2} 

Since the collision is elastic, so the K.E of the system before and after collision is conserved .  


Thus  
½ m_{1}v_{1}^{2} + ½ m_{2}v_{2}^{2} = ½ m_{1}u_{1}^{2} + ½ m_{2}u_{2}^{2}
½ (m_{1}v_{1}^{2} + m_{2}v_{2}^{2}) = ½ (m_{1}u_{1}^{2} + ½ m_{2}u_{2}^{2} m_{1}v_{1}^{2}m_{1}u_{1}^{2}=m_{2}u_{2}^{2}m_{2}v_{2}^{2} m_{1}(v_{1}^{2}u_{1}^{2}) = m_{2}(u_{2}^{2}v_{2}^{2}) m_{1}(v_{1}+u_{1}) (v_{1}u_{1}) = m_{2}(u_{2}+v_{2}) (u_{2}v_{2}) —— (2) 

Dividing equation (2) by equation (1)  


V_{1}+U_{1} = U_{2}+V_{2 }


From the above equation  
V_{1}=U_{2 }+V_{2 }U_{1}_________(a)
V_{2}=V_{1}+U_{1 }U_{2}_________(b)


Putting the value of V_{2} in equation (1)  
m_{1} (v_{1}u_{1}) =m_{2} (u_{2}v_{2})
m_{1} (v_{1}u_{1}) =m_{2}{u2(v_{1}+u_{1}u_{2})} m_{1}(v_{1}u_{1})=m_{2}{u_{2}v_{1}u_{1}+u_{2}} m_{1}(v_{1}u_{1})=m_{2}{2u_{2}v_{1}u_{1}} m_{1}v_{1}m_{1}u_{1}=2m_{2}u_{2}m_{2}v_{1}m_{2}u_{1} m_{1}v_{1}+m_{2}v_{1}=m_{1}u_{1}m_{2}u_{1}+2m_{2}u_{2} v_{1}(m_{1}+m_{2})=(m_{1}m_{2})u_{1}2m_{2}u_{2} 



In order to obtain V_{2} putting the value of V_{1} from equation (a) in equation (i)  
m_{1 }(v_{1}u_{1}) = m_{2}(u_{2}v_{2})


m_{1}(u_{2}+v_{2}u_{1}u_{1})=m_{2}(u_{2}v_{2})
m_{1}(u_{2}+v_{2}2u_{1})=m_{2}(u_{2}v_{2}) m_{1}u_{2}+m_{1}v_{2}2m_{1}u_{1}=m_{2}u_{2}m_{2}v_{2} m_{1}v_{2}+m_{2}v_{2}=2m_{1}u_{1}+m_{2}u_{2}m_{1}u_{2} v_{2}(m_{1}+m_{2})=2m_{1}u_{1}+(m_{2}m_{1})u_{2} 

