 Solution

The zero initial displacement selects sine time factors. Put ${\omega }_1=\pi c/L$. The Fourier sine series of the initial velocity has coefficients $V_1=V_0/2$, $V_2=-3V_0/4$, with all other coefficients zero. Since differentiation in time multiplies a sine by its angular frequency,The normal mode coordinates are $q_1(t)=V_0\sin ({\omega }_{1}t)/(2{\omega }_1)$ and $q_2(t)=-3V_0\sin (2{\omega }_{1}t)/(8{\omega }_1)$. Their peak displacement amplitudes are $|V_0|/(2{\omega }_1)$ and $3|V_0|/(8{\omega }_1)$. Both the initial displacement and the initial velocity can be checked directly from this expression.