Dumbbell and Spring

A spring dumbbell comprises two balls of mass $m$ that are connected with a spring of stiffness $k$ . Two such dumbbells are sliding towards one another on a smooth horizontal surface, the velocity of either is $v_{0}$ . At some point, the distance between them is $L$ .

The time in which the distance between them $L$ again can be expressed as $\frac{\alpha L}{v_{0}}+\beta \sqrt{\frac{m}{k}} .$

What is the sum of constants, $\alpha + \beta$ ?

Assume the collisions are perfectly elastic.