are on the dividing line of stop and roll-out. Some of the ricochets bounced twice and are labelled with a red diamond. Each point represents a single impact trial using the pendulum launcher. \[fig:ric\] ](ric.png){width="3.3in"}

The critical angle for ricochet {#sec:crit}
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A ricochet takes place when the lift force is at least large enough to overcome the gravitational force and this must happen before drag forces reduce the horizontal velocity component velocity to zero [@johnson75; @soliman76; @bai81]. For spherical projectiles, a lift force dependent on the square of the depth times the square of the velocity and a constant downward gravitational acceleration were adopted by @soliman76 to estimate a critical impact angle for ricochet $$\theta_{cr}^2 \sim \frac{1}{10} \frac{\rho_s}{\rho_p} - \frac{4 R_p g}{v_{impact}^2} .
 \label{eqn:soliman}$$ At a particular velocity, ricochets occur at impact angles below the critical one. The term on the right is proportional to the inverse of the square of the Froude number and was originally calculated for ricochets on water [@johnson75; @birkhoff44]. The term on the left is only dependent on the substrate and projectile density ratio. At high velocities the critical angle only depends on the density ratio. In the limit of high velocity, equation \[eqn:soliman\] 