## Physics XI Content

# ELASTIC AND INELASTIC COLLISION

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 non-rotating 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} | ||