Maths Question.

[bobalot]

Is it possible to get a distance set a over period of time from a acceleration vs velocity graph?

The graph looks like this:



If I had the numbers for acceleration and velocity for that graph, is it actually possible to get a distance over lets say 30 seconds?

[CaptainZoidberg]

Do you have the actual function for that graph (as opposed to just the numbers). If so, you just have a differential equation:

dv/dt = f(v)

which you could solve by hand with laplace transforms, or with a TI-89 or maple

If you just have the data, then I think that you can solve the problem with eigenvectors, where you basically represent your differential equation

v' = f(v)

as:

Av = cv

where A is the matrix that has your data in it, and c is a constant that you compute as you solve the problem. I'm a little sketchy on the details of this method...

[Jaepheth]

If you have a function A(v) that gives the acceleration for any given speed v, and an initial speed at time = 0, then You can probably convert it to a speed vs time graph and get your distance traveled from that.

[bobalot]

Do you have the actual function for that graph (as opposed to just the numbers). If so, you just have a differential equation:

dv/dt = f(v)

which you could solve by hand with laplace transforms, or with a TI-89 or maple

If you just have the data, then I think that you can solve the problem with eigenvectors, where you basically represent your differential equation

v' = f(v)

as:

Av = cv

where A is the matrix that has your data in it, and c is a constant that you compute as you solve the problem. I'm a little sketchy on the details of this method...

Unfortunately I only have the data. It's actual field data. I do remember how to do laplace transforms, I have really forgotten matrices, eigen vectors and all that stuff. I will have to dig up my old notes on it.

[Kuroneko]

Assuming dv/dt = A(v), then since dv/dt = [dx/dt][dv/dx] = v dv/dx, substition v dv/A(v) = dx, and therefore x = x₀ + Int_{v₀}^{v}[ ν dν/A(ν) ]. On the other hand, integrating without the substition gives t = t₀ + Int_{v₀}^{v}[ dν/A(ν) ], which for any particular t one can numerically solve for the velocity.

[CaptainZoidberg]

Hmm... Nice work Kuroneko. Do you by any chance know what the name is for that method?

[Steel]

The key bit of this is the a=vdv/dx substitution. That's a pretty standard thing in dynamics, I don't think it is important enough to have a name. The rest is just rearranging.