I am building a spring launcher for an engineering project and need to calculate spring compression to achieve a projectile velocity of $4$ m/s. The mass $m_p$ of the projectile is $5$ g, the mass of the spring is $7$ g and the spring constant is $1000$ N/m.
Is it right to assume that initial potential energy in the spring = kinetic energy of the marble + kinetic energy of the spring at the point of release?
I can calculate the marble's kinetic energy easily enough using $E_k=\frac12m_pv^2$. However, how would I calculate the spring's kinetic energy? One end is secured, while the other end would be traveling at the same speed as the marble. I think I would have to use integration somehow but not completely sure how to.