I am trying to plot the speed of an object of known mass, area and drag coefficient in a given density of fluid.
I have correctly calculated the terminal velocity by
However I have tried a few methods to determine the velocity between t=0 and a given time. They all involve gravity and mass for the force and when I swap out for a given force, it all goes wrong.
Hopefully somebody can assist me.
Some more info, the object (a boat) is initially at rest before a constant thrust force is applied. $$f = ma$$ rearranged gives acceleration and applying delta-t gives me the speed at any given time, however it all goes wrong when i apply a drag equation,
$$F_d = [0.5 * (Rho * V^2)] * [Cd] * [Area]$$
I can correctly calculate the increase in drag with speed and initially tried an iterative formula in excel which solved the applied force minus the drag force to give net force and thus acceleration, applying a dT step and using that for V, and having excel loop until a steady V was found where Fd balanced the applied thrust. It worked to a point however it never successfully found a point which matched a terminal velocity calculation, which i was using to check the incremental time step calculation.
The closest i have got is using a free falling object calculation in a spreadsheet called "Falling Motion under Gravity: Resistance as Velocity Squared" by Michael Fowler, University of Virginia:
$$V_t = [Vprev] + (g-((Cd/m)*[Vprev]^2)*deltaT$$
But (and in answer to a question) my calculus is no where near good enough to put all of this into a pot and come up with a way to remove gravity and add in the Rho of water and drag area of the boat. We are ignoring air as the V will be low, less than 3m/sec so air resistance is going to be negligible.
From the terminal velocity calculation I get a sensible value for the final speed, i just want to be able to plot speed with regards to time and end up with a flat line representing the terminal velocity.
Many thanks :)