Relation between coefficient of restitution for perfectly elastic collision and perfectly inelastic collision
Answers
In a one-dimensional collision, the two key principles are: conservation of energy (conservation of kinetic energy if the collision is perfectly elastic) and conservation of (linear) momentum. A third equation can be derived[6] from these two, which is the restitution equation as stated above. When solving problems, any two of the three equations can be used. The advantage of using the restitution equation is that it sometimes provides a more convenient way to approach the problem.
Let {\displaystyle m_{1}} m_{1}, {\displaystyle m_{2}} m_{2} be the mass of object 1 and object 2 respectively. Let {\displaystyle u_{1}} u_{1}, {\displaystyle u_{2}} u_{2} be the initial velocity of object 1 and object 2 respectively. Let {\displaystyle v_{1}} v_{1}, {\displaystyle v_{2}} v_{2} be the final velocity of object 1 and object 2 respectively.
{\displaystyle {\begin{cases}{\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}\\m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}\end{cases}}} {\displaystyle {\begin{cases}{\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}\\m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}\end{cases}}}
From the first equation,
{\displaystyle m_{1}(u_{1}^{2}-v_{1}^{2})=m_{2}(v_{2}^{2}-u_{2}^{2})} {\displaystyle m_{1}(u_{1}^{2}-v_{1}^{2})=m_{2}(v_{2}^{2}-u_{2}^{2})}
{\displaystyle m_{1}(u_{1}+v_{1})(u_{1}-v_{1})=m_{2}(v_{2}+u_{2})(v_{2}-u_{2})} {\displaystyle m_{1}(u_{1}+v_{1})(u_{1}-v_{1})=m_{2}(v_{2}+u_{2})(v_{2}-u_{2})}
From the second equation,
{\displaystyle m_{1}(u_{1}-v_{1})=m_{2}(v_{2}-u_{2})} {\displaystyle m_{1}(u_{1}-v_{1})=m_{2}(v_{2}-u_{2})}
After division,
{\displaystyle u_{1}+v_{1}=v_{2}+u_{2}} {\displaystyle u_{1}+v_{1}=v_{2}+u_{2}}
{\displaystyle u_{1}-u_{2}=-(v_{1}-v_{2})} {\displaystyle u_{1}-u_{2}=-(v_{1}-v_{2})}
{\displaystyle -{\frac {v_{1}-v_{2}}{u_{1}-u_{2}}}=1} {\displaystyle -{\frac {v_{1}-v_{2}}{u_{1}-u_{2}}}=1}