# equation for velocity vs displacement for nonlinear acceleration [closed]

An object starts out at t=0, s=0 and accelerates with A=sin(pi*t/T), where T is defined as (pi/2 * Vmax/Amax). Vmax is maximum velocity, and Amax is maximum acceleration. Solution is restricted to t=0 to t=T

The velocity equation is v(t) = 0.5Vmax(1-cos(pi*t/T))

What I am trying to solve is velocity as a function of displacement, v(s).

In other words, without having access to elapsed time, I am trying to generate an equation for velocity as a function of the observed displacement.

Is this possible?

Thanks

## closed as off-topic by Danu, Daniel Griscom, Gert, user36790, John RennieJan 25 '16 at 7:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Danu, Daniel Griscom, Gert, Community, John Rennie
If this question can be reworded to fit the rules in the help center, please edit the question.

Given the acceleration $a = \sin \left( \frac{\pi t}{T}\right)$, by integration you get

$$v(t) = \int_0^t a\,{\rm d}t = \frac{T}{2} \left(1-\cos\left(\frac{\pi t}{T}\right) \right)$$ $$s(t) = \int_0^t v\,{\rm d}t = \frac{T}{\pi^2} \left(\pi t - T \sin\left(\frac{\pi t}{T}\right) \right)$$

Since the last one cannot be inverted for $t(s)$, we can invert the first one to get $t(v)$ and use it in the second one

$$t(v) = \frac{T}{\pi} \cos^{-1} \left( 1- \frac{\pi v}{T} \right)$$

$$s(v) = \frac{T^2}{\pi^2} \cos^{-1} \left( 1- \frac{\pi v}{T} \right) - T \sqrt{ \frac{2 T v}{\pi^3} - \frac{v^2}{\pi^2} }$$

• The purpose of my query is an experimental attempt to operate a track mounted vehicle (powered by a variable frequency drive) which has a position measuring transducer, and to guide it to a stop at a specific position. I have already succeeded controlling it using constant acceleration, using v=SQRT(2AS). I am attempting to improve by reducing the jerk in the system with the acceleration I proposed. I had a typo in my original question in that the acceleration is Amax * sin(pi * t/T). However I need v(s), not s(v). I recognize my calculus is rusty. Thanks – MartyH Jan 25 '16 at 20:31
• $v(s)$ does not exists analytically. This is because $s(t)$ cannot be solved for $t$ since it is of the form $t=a \sin(t)$. Otherwise you could state $v(s) = v(t(s))$ from $v(t)$. – ja72 Jan 25 '16 at 20:37
• Thanks for the insight. Based on your input, I instead created a spreadsheet table of normalized s and v values. I can now accomplish the goal using a table lookup. – MartyH Jan 26 '16 at 21:28
• So my answer was useful. Great. – ja72 Jan 27 '16 at 3:15