In the annual boat race between Oxford and Cambridge the drag force F experience

Question

In the annual boat race between Oxford and Cambridge the drag force F experienced
by a boat is proportional to S1/4 v where S is the surface area of the hull of the boat and v
is the velocity of the boat. From physics we know that the power P required to overcome F
is given by P = Fv. Further, suppose that P is proportional to n, the number of boatmen
on board and n is proportional to the length l of the boat. Find the relationship between
v and n assuming the boats are geometrically similar. (Hint: Use geometric similarity and
let l be the characteristic dimension where S = l^2.)

Explanation / Answer

1) We are seeking the function v(n) which relates the velocity of a boat to the number of towers. 2) Given the relationship P=vF, we can rearrange as v=P/F 3) Now P is proportional to n and F is proportional to v S^(1/4) or to v(l^2)^(1/4). 4) Since l is proportional to n we can substitute n for l in step 3 to obtain the conclusion that F is proportional to v(n^(1/2)) 5) Substituting from steps 3 and 4 into step 2 we obtain: v= kn/(vn^(1/2)) =k n^(1/2)/v where k is some constant 6) Rearranging: v^2 = k n^(1/2) or v is proportional to n^(1/4), that is v is proportional to the fourth root of n

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