The general notion might be from physics, but I am very much interested in the math core of the problem. I've been trying to figure it out for a while, with little success.
The problem states:
What is the height from which an object was dropped if it travelled the last $x$ units of distance in $t_x$ units of time?
Specific values and units are not important. It's just a thought problem, for the sake of it. So, we don't have any literals, but what we do have is:
The total height would be $h_0$ and total time would be $t_0$. The equation for the height can be obtained as an indefinite integration of the function $v(t)$ or through more common channels which do the same thing.
$h_0 = \frac{gt_0^2}{2}$ (1)
Right, that much is clear. This much is also true:
$h_0 = h + x$ (2)
$t_0 = t + t_x$ (3)
$h = \frac{gt^2}{2}$
Which enables us to restate the equation (1) as:
$h+x= \frac{g}{2}(t+t_x)^2$
Now, we know the values of $x$ and $t_x$ and the value of gravitational acceleration, $g= ~9.80665$ $m/s^2$
All that remains is $h$ and $t$ and I just can't express it, everything I try to do doesn't give me an insight into their values. Is the system of equations under-constrained? I would really appreciate some insight, even if it is just to show the error of my ways.
So, the point of all this is, how can I obtain the original height function just from knowing two facts, the length traveled and the time it took to do it.
Can a function be reconstructed? And how would one go about doing it? Without succumbing to averaging the end velocity (stating $x/t_x$) and then figuring out how it "decayed" backwards in time, therefore reaching an approximation of the height.