A sailboat of mass m is driven by a constant wind force, so…

A sailboat of mass $m$ is driven by a constant wind force, so that its position is $s = At^2 + Bt + C$ ($A$, $B$, $C$ constants). Find (a) the work done by the wind during the time $t$ from the start, (b) the instantaneous power of the wind. (Water resistance is neglected.)

Explanation of Method

Work and power of the wind on a sailboat.

Step 1: Velocity, acceleration, force

\[ v = 2At + B, \]

\[ a = 2A, \]

\[ F = 2mA. \]

Step 2: (a) Work

\[ W = F\,(s(t) - s(0)) = 2mA\,(At^2 + Bt). \]

\[W = 2mA\,(At^2 + Bt)\]

Step 3: (b) Power

\[P = Fv = 2mA\,(2At + B)\]

Check

$ \Delta E_k = \tfrac12 m\big[(2At + B)^2 - B^2\big] = 2mA(At^2 + Bt) = W\ \checkmark $

$v$ $t$ $B$ $t$ $v = 2At + B$ $s - C$ $F = 2mA$ $W = \Delta E_k = 2mA(At^2 + Bt)$

The speed grows linearly from $B$, so the wind's force $2mA$ is constant; its work is force times the distance covered (the shaded area).

Result

\[\boxed{\,W = 2mA\,(At^2 + Bt)\,}\]

\[\boxed{\,P = Fv = 2mA\,(2At + B)\,}\]

Remark

Check. The original results are correct.