Ship Scale$-$up Experiments with scale models are sometimes used to guide ship design. One quantity of interest is the drag force $($FD$)$ that opposes the ship$$s forward motion. It can be evaluated for a model by towing at a steady speed in a test tank and measuring the tension on the tow line. It is reasonable to suppose that FD depends on the ship speed U$,$ ship length L$,$ k other ship dimensions $($d$1,$ d$2,...,$ dk$),$ and p of water, and g$.$ Gravity has role because surface ships are affected by waves, including the ones they generate. You may assume that the model size greatly exceeds the capillary length, making surface tension negligible.

$($a$)$ What can you conclude about FD from dimensional analysis?

$($b$)$ The largest practical model of a modern ship might be $1/100$ scale, with a length of

about $1$ m$.$ Ideally, the model in a towing tank would be dynamically similar to the

full$-$size ship. To what extent can that be achieved?