Time resolved measurements of the growth of Raman instabilities were performed using a picosecond chirped laser pulse. It was observed experimentally that for a short laser pulse (<10 ps), forward and 30 degrees Raman scattering occur at the back of the pulse. The growth of the instabilities was found to be independent of the sign of the chirp. In addition, a simple temporal model was developed and shows good agreement with the experimental results. This model also indicates that the plasma wave driven by forward Raman scattering is severely damped in the case of pulses longer than a few picoseconds. Damping by the modulational instability is compatible with the experimental results.