Question #79715

A micro-submarine navigates in a 2D fluid flow. The fluid velocity field is given by $\vec{v}_f = -\nabla \Phi$, where the scalar potential is $\Phi(x,y) = k(x^2 - y^2)$ and $k$ is a positive constant with units of $\text{s}^{-1}$. The submarine moves from point $A(R, 0)$ to point $B(0, R)$ along a quarter-circular path $x^2 + y^2 = R^2$ in the first quadrant, where $R$ is in meters. The submarine's absolute velocity $\vec{v}$ (velocity with respect to the stationary ground) is maintained such that its magnitude is a constant $v_0$ (in $\text{m/s}$) and it is always directed tangentially along the path from $A$ to $B$. The fluid exerts a drag force on the submarine given by $\vec{F}_d = -b \vec{v}_r$, where $b$ is a positive constant with units of $\text{kg} \cdot \text{s}^{-1}$ and $\vec{v}_r$ is the velocity of the submarine relative to the fluid. The submarine's internal engine provides a thrust $\vec{F}_t$ to exactly balance the drag force at all times (assume no other forces act in the plane of motion). Calculate the total work $W_t$ done by the engine as the submarine moves from $A$ to $B$.