How long to travel fixed distance with variable velocity?

Question

Suppose we have an underwater robot that can travel at a fixed velocity, v, if there is no current. This robot needs to travel a fixed distance, d. While in the water, the current acts on the robot affecting its velocity. The current varies over time. How do we determine how long it takes for the robot to travel d? That is, can we get a general equation for time given d and v?

Answer 1

Total speed: $$v_t= c(t)+v_r$$ where $c$ is the function of the current and $v_r$ is the fixed velocity you mentioned.

Integrating: $$\int_0^x(c(t)+v)dt = \int_0^x v dt + \int_0^x c(t) dt = vx + C(x)-C(0) + U$$ Where $x$ is the time it would take for the robot to cover the distance $d$, $C(x)$ is the integral of $c(x)$, and $U$ is the integration constant.

Now solving for x: $$ vx + C(x) - C(0) = d$$ It is quite obvious that the solution of the equation vastly depends on $c(t)$.

Note: $U = 0$. This is the general solution.