Displacement vs time under a time varying speed limit

[Stephen Tashi]

TL;DR Summary

The speed limits on a straight road are given by a known function g(x,t) where x is the location on the road and t is time. A car starts at x = 0 at time t= 0 and always drives at the speed limit. The location of the car is given by the (unknown) function s(t). Is there a differential equation that defines s(t)?

The speed limits on a straight road are given by a known function g(x,t) where x is the location on the road and t is time. A car starts at x = 0 at time t= 0 and always drives at the speed limit. The location of the car is given by the (unknown) function s(t). Is there a differential equation that defines s(t)? Assume g(x,t) is differentiable function (unlike the way real speed limit laws are).

(The motivation for the problem is wondering about how to model a "disturbance" propagating through a medium where the properties of the medium determine the velocity at which the disturbance propagates - but the version using a car and speed limits sounds more concrete.)

If $s(t)$ is 1-to-1 then we have $s'(t) = g( s^{-1}(t),t)$. That could be called a differential equation, but is there a more usual type of differential equation for it?

 

[hutchphd]

If $s(t)$ is 1-to-1 then we have $s'(t) = g( s^{-1}(t),t)$. That could be called a differential equation, but is there a more usual type of differential equation for it?

Why is $x=s^{-1}(t)$??

 

[Stephen Tashi]

Why is $x=s^{-1}(t)$??

Why indeed! I'm wrong. It should be $s'(t) = g(s(t),t)$.

I should also clarify that the speed limit $g(x,t)$ may be a negative number.

 

[PeroK]

Why indeed! I'm wrong. It should be $s'(t) = g(s(t),t)$.

This is called a first order ordinary differential equation. If the equation is linear, i.e. of the form: $$s'(t) = sf(t) + h(t)$$ Then it can be solved in general using an integrating factor.

If the equation is non-linear, then there is no general analytic method of solution. See, for example:

https://www-thphys.physics.ox.ac.uk/people/FrancescoHautmann/Cp4/sl_ode_11_2.pdf

 

[pasmith]

Summary:: The speed limits on a straight road are given by a known function g(x,t) where x is the location on the road and t is time. A car starts at x = 0 at time t= 0 and always drives at the speed limit. The location of the car is given by the (unknown) function s(t). Is there a differential equation that defines s(t)?

The speed limits on a straight road are given by a known function g(x,t) where x is the location on the road and t is time. A car starts at x = 0 at time t= 0 and always drives at the speed limit. The location of the car is given by the (unknown) function s(t). Is there a differential equation that defines s(t)? Assume g(x,t) is differentiable function (unlike the way real speed limit laws are).

(The motivation for the problem is wondering about how to model a "disturbance" propagating through a medium where the properties of the medium determine the velocity at which the disturbance propagates - but the version using a car and speed limits sounds more concrete.)

This sounds like a problem in continuum mechanics.

Why indeed! I'm wrong. It should be $s'(t) = g(s(t),t)$.

I should also clarify that the speed limit $g(x,t)$ may be a negative number.

This result will hold even if $g$ is negative.

Indeed in fluid mechanics one has the local fluid velocity $\mathbf{u}(\mathbf{x},t)$ which is found by solving a system of PDEs and the trajectory of a particular particle can then be found by solving $$\frac{d \mathbf{X}}{dt} = \mathbf{u}(\mathbf{X}(t),t).$$

 

[Office_Shredder]

Does the speed limit really depend on time? Given the setup in the op, you can probably write this as
$\frac{ds}{dt} = g(s)$
And then you can solve it using separation of variables.

 

[Stephen Tashi]

Does the speed limit really depend on time?

Yes, the way I'm thinking of the problem. For example, with real speed limits, there are signs on streets near schools that say "25 MPH when flashing" .

 

[Office_Shredder]

In that case I don't think there is going to be a nice closed form solution that's easy to write down. If g is piecewise constant then you know that s is piecewise linear, and computing all the pieces and gluing them together doesn't seem to be that hard to me.