Analytical solutions for the initial rise height $z_m$ of a turbulent fountain for the limits of both small and large source Froude number $\hbox{\it Fr}_0$ are presented. These solutions are based on a plume entrainment model. For large Froude number fountains, the established result $z_m/r_0{\,\sim\,}\hbox{\it Fr}_0$ is obtained ($r_0$ denoting the source radius). For intermediate Froude numbers, the relationship $z_m/r_0{\,\sim\,}\hbox{\it Fr}_0^2$ is found and the rise height is independent of the entrainment coefficient $\alpha$. For very small Froude numbers, the flow is hydraulically controlled at the source and $z_m/r_0{\,\sim\,}\hbox{\it Fr}_0^{2/3}$. Existing experimental and numerical results, as well as our own experimental results, are compared to our solutions and show good agreement. Comparison with experimental results also demonstrates that the appropriate entrainment coefficient for highly forced fountains is $\alpha_f{\,\approx\,}0.058$. This is significantly closer to the entrainment coefficient of a jet than of a plume.