predicts a critical angle of $15^\circ$ for our substrate to projectile density ratio of $\rho_s/\rho_p \sim 0.66$. At the higher velocities in Figure \[fig:ric\], we saw ricochets at impact angles up to $40^\circ$, so this model does not apply very well in the regime of our experiments.

Equation \[eqn:soliman\] predicts that the critical angle is larger at higher velocity. This behavior is seen on water, but ricochets on sand can deviate from this behavior and can scale in the opposite way with critical angle decreasing at higher velocity (see Figure 8 by @soliman76). @bai81 modified equation \[eqn:soliman\] with the addition of a constant pressure term dependent on parameter $K'$ adding to the lift, $$\theta_{cr}^2 \sim \frac{1}{10} \frac{\rho_s}{\rho_p} - \frac{4 R_p g}{v_{impact}^2}  + \frac{K'}{v_{impact}^2}.
 \label{eqn:bai}$$ This additional term allowed them to account for a decreasing critical angle with increasing impact velocity, which was seen in their experiments.

As we see an increase in impact angle with increasing velocity, the simpler model by @soliman76 might give a line that matches the division seen in our experiments. We found a similar line that does delineate the impact outcomes and it is shown in Figure \[fig:ric\_flip2\] as a dotted orange line. Figure \[fig:ric\_flip2\] shows 