# Speed as a function of time

1. Aug 7, 2011

### Matej

How could I express the speed of an object as a function of time if the object is accelerating with a constant power output and is being affected by quadratic (Rayleigh) drag?

2. Aug 7, 2011

### SteamKing

Staff Emeritus
What type of object?

3. Aug 7, 2011

### Matej

Well, it's supposed to be a submarine.
As far as the physics go, it's a solid object with density equal to the density of the liquid
Anything else?

4. Aug 7, 2011

### SteamKing

Staff Emeritus
As a start, I would draw a free-body diagram and add the relevant forces. Since it is a submarine, the power produced by the propulsion machinery is converted into thrust, either by turning a propeller or by means more exotic. The drag is proportional to the velocity squared and acts to oppose the thrust. Since the sub is accelerating, then T - D is positive, and T - D = m a, where m is the mass of the submarine, and a = dv/dt.

5. Aug 7, 2011

### A.T.

And in general the thurst will not be constant for constant power input, but depend on the speed.

6. Aug 8, 2011

### Matej

And that's the point where I got stuck. I found an equation that works if the thrust is constant but that is not the case here.

Perhaps if I expressed drag as a loss in energy (speed3) it might be simpler.

Not quite sure though, we haven't done anything like this at school so far and it might be a few years till we do.

7. Aug 8, 2011

### A.T.

You won't get an analytic solution for this. The thrust itself is not some simple function but usually derived empirically or numerically. You have a chart like this:

And get the thrust by:

thrust = (efficiency * shaft_power) / velocity

To get the speed as function of time you have to integrate the acceleration from the net force (thrust - drag) numerically. For low velocities there is static thrust data.

Last edited: Aug 8, 2011