# Integral of speed as a function of displacement

Hello,

I'm trying to figure out a way to do an integral of a speed/velocity function v(x) described as a function of displacement (the speed changes based on the position of the particle). I want to know the distance traveled after a specific period of time, and I can't figure out how to formulate the integral for that.

Do path integrals or integration by parts play a role in this? Thanks.

haruspex
Homework Helper
Gold Member
2020 Award
You have dx/dt = f(x)?
Turn that into dt = dx/f(x) and integrate, if you can:

t = $\int$(1/f(x)).dx

However, you mentioned path integrals, so maybe there's more than one space dimension here?

I don't have a specific problem to show, just an idea I want to follow up on.

It may be true that there are no path integrals (as of yet) as you mention involved in the problem, just one dimension so far. But I mention it just in case the solution may require expansion of the problem into more dimensions to find the solution.

I've tried the integrating for t that you show, but maybe I'm not stating the problem correctly. I want to find the solution after a specific time t1 of the distance traveled, given the speed (one dimension) as it changes based on position x, so the displacement x' depends on position x and time t.

I guess I'm looking for something in the form:

x' = f(v(x),t1),

and I'm sure it takes at least some form of integral. The velocity is a function of time only so far as it changes as it crosses space, and the position in space changes the velocity. Would it make sense to have a function of space as it changes over time, maybe stretching the space variable while keeping velocity constant with respect to a stretching space? Maybe I have to generate acceleration functions before going back to derive displacement? Or is there a simple solution I haven't remembered from my college days?

There must be some answer to something like this already, though I have been searching online and haven't found it yet, unless it's more complex than I'm familiar with. I'm thinking of solutions to a particle moving through a space-dependent force field.

haruspex
Homework Helper
Gold Member
2020 Award
As I understand it, you have speed v = v(x), so
t = ∫(1/v(x)).dx
If you want to know the position x1 reached at time t1 then you have to solve
t1 = $∫^{x1}_{0}$(1/v(x)).dx

E.g. suppose v(x) = a.x + b
t1 = [ln(x+b/a)/a$]^{x1}_{0}$ = (ln(x1+b/a) - ln(b/a)]/a
a.x1 = b.exp(a.t1) - b

If that's not what you're trying to do, please pick a specific v=v(x) so that we an discuss it more clearly.

As I understand it, you have speed v = v(x), so
t = ∫(1/v(x)).dx
If you want to know the position x1 reached at time t1 then you have to solve
t1 = $∫^{x1}_{0}$(1/v(x)).dx

E.g. suppose v(x) = a.x + b
t1 = [ln(x+b/a)/a$]^{x1}_{0}$ = (ln(x1+b/a) - ln(b/a)]/a
a.x1 = b.exp(a.t1) - b

If that's not what you're trying to do, please pick a specific v=v(x) so that we an discuss it more clearly.

Thank you, it's kind of what I was looking for; basically, as you confirmed, it's an inverse problem where the variable that is to be solved is the boundary of the integral. I'll work on this a bit and see what I come up with.