Finding Position from Equations of Motion at Different Times

[PFuser1232]

Usually, equations of motion (with constant acceleration) are written in terms of values of position/velocity at time $t=0$.
Take for example:

$$x = x_0 + v_0t + \frac{1}{2} a t^2$$

Where $x_0$ and $v_0$ are the values (at $t=0$) of position and velocity respectively.

What if we're given position at some other time $t=t_0$, instead of $t=0$, and we're asked to find $x$ as a function of time?

What I do is I find a more general form of the equation I wrote above as follows:

$$\int_{x_0}^x dx' = \int_{t_0}^t v(t') dt'$$

Where $x_0$ now represents position at time $t=t_0$.

Another way would be to eliminate the variable $x(0)$

Is there any alternative approach?

 

[Chestermiller]

This approach is great. Of course, what you would find is that you could also just substitute (t - t₀) for t in the original equation (i.e., shift the time scale), assuming that v₀ is the velocity at t = t₀.

Chet