The falling droplet technique is used to determine the extensional properties of very dilute and semi-dilute polymeric solutions. Nine different aqueous solutions of Polyacrilamide E10 were tested spanning a wide range of polymer concentration. The extensional viscosity for all solutions was found to be orders of magnitude larger than the shear viscosity. The extensional viscosity increases rapidly with time after the start of stretching. It was also shown that there is a linear relationship between the stress and the strain in the fluid filament, indicating that the fluid behaves like an elastic solid. The modulus of elasticity of the fluid was found to increase rapidly at low polymer concentrations, up to 0.1%, with no subsequent increase at higher concentrations. The stretch rate was found to be a decreasing function of polymer concentration. The stretch rate determined from the measurement of filament diameter was found to be greater than that determined by measurement of filament length. The spread between the two experimentally determined stretch rates was found to decrease with polymer concentration. This discrepancy is due to the small forward flow from the filament into the droplets and the back flow from the filament into the filament attachment region due to capillary action.