While working with problems on elastic collisions, I have come across this observation, that the elastic potential energy of a two-body system is the maximum when the relative velocity equals zero. In fact, the same applies when we are talking about a 2-block+spring system. Why is it so?
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asked Jan 5, 2021 at 15:25
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In case you need some mathematical insight,
Nothing specific to add but when you calculate the kinetic energy of a system(I will assume it to be a 2 body system but you can generalize) the Kinetic Energy at a given point of the system is given by: $$ \frac{1}{2}{m_{total}}V_{CM}^2 + \frac{1}{2}{\mu}{V_{rel}^2} $$
where $\mu$ is reduced mass: $\frac{m_1m_2}{m_1 + m_2}$
As total energy is constant you need to minimize kinetic energy to maximize potential energy, as $V_{CM}$ is constant, PE is max when $V_{rel}$ is 0.
Or if you visualize the motion you may think that initially when the bodies collide, compression will take place -> They will move at equal velocity(max compression) -> the spring will regain its natural length.
