Prob. $7\mathrm{.}50[$Ans: $1\mathrm{.}9f\frac{t}{s}].$

$7\mathrm{.}50($See Fluids in the News article titled "Ice engineering," Section $7\mathrm{.}9\mathrm{.}3.)$ A model study is to be developed to determine the force exerted on bridge piers due to floating chunks of ice in a river. The piers of interest have square cross sections. Assume that the force, $R,$ is a function of the pier width, $b,$ the thickness of the ice, $d,$ the velocity of the ice, $V,$ the acceleration of gravity, $g,$ the density of the ice, ${}_i,$ and a measure of the strength of the ice, $E_i,$ where $E_i$ has the dimensions $F{L}^{-2}.($a$)$ Based on these variables determine a suitable set of dimensionless variables for this problem. $($b$)$ The prototype conditions of interest include an ice thickness of $12in.$ and an ice velocity of $6f\frac{t}{s}.$ What model ice thickness and velocity would be required if the length scale is to be $\frac{1}{10}?($c$)$ If the model and prototype ice have the same density, can the model ice have the same strength properties as that of the prototype ice? Explain.