Position-time equation from force-position equation

[Droctagonopus]

I've been trying to obtain an equation of position in terms of time given force in terms of position. I've tried and I think I've managed to obtain an equation of velocity in terms of position using work and kinetic energy but I haven't managed the position time equation.

This is how I got velocity equation:
A point mass m is at starting position x=0 with starting velocity v_0

F=x+1
W=\int_0^x {(x+1)dx}=\frac{x^2+2x}2
\frac 1 2 m(v^2-{v_0}^2)=\frac{x^2+2x}2
v=\sqrt{\frac{x^2+2x}m+{v_0}^2}

How do I get a position-time equation? And how do I use a starting position other than x = 0?

 

[WannabeNewton]

Well just note that your equations are horribly dimensionally incorrect. But other than that, if you have $\dot{x} = f(x)$ then $t = \int _{0}^{x} \frac{1}{f(x)}dx$ and from there you just have to integrate (in principle) and invert the equation (in principle) to get $x(t)$ explicitly.

 

[Droctagonopus]

Sorry I forgot to mention that I was working in one dimension. And what does the dot above the x signify?

 

[WannabeNewton]

When I said dimensionally incorrect I meant that your units are all wrong. The dot is a time derivative so $v = \dot{x}$.

 

[Droctagonopus]

By inverting the equation, do you mean getting an equation of x in terms of t?

 

[WannabeNewton]

Yep.

 

[Droctagonopus]

But does the velocity equation have to have a time variable in it? And also, is it possible to do this from an acceleration equation?

 

[WannabeNewton]

No the velocity equation doesn't have to have a time variable. As I said, you have $\frac{\mathrm{d} x}{\mathrm{d} t} = f(x)$ hence you can solve for $t(x)$ just by solving the differential equation (in principle). You can then get $x(t)$ by inverting the equation (again in principle). I don't know what you mean by your second question; isn't that exactly what you did here? You started with the force as a function of position, which is also acceleration as a function of position, and you used the work-energy theorem to get an equation for the velocity in terms of position.

 

[Droctagonopus]

I mean, is there was a way to do it without going through the process of obtaining a velocity equation?

 

[WannabeNewton]

Well sure you could directly solve the second order linear differential equation in principle. The simplest example is the simple harmonic oscillator wherein you have $\ddot{x} = f(x) = -\omega^{2} x$. You can solve this analytically immediately and obtain $x(t) = A\cos\omega t + B\sin\omega t$.