I have a list of accelerations and time intervals, and need to calculate the total distance travelled at each point in the list ($v_0 = 0 m/s$ and $d_0 = 0 m$).
The standard formula looks like this:
$$d = v_0 \cdot t + \frac{a \cdot t^2}{2}$$
However, I have found that I can calculate the cumulative velocity and from that the cumulative distance with these formulas (I'm actually doing it in Excel and just adding the value in the previous row, but these formulas should reflect the approach I'm using):
$v_n = v_{n-1} + a_n \cdot t_n$ for $n>0$ and $v_0 = 0$
$d_n = d_{n-1} + v_n \cdot t_n$ for $n>0$ and $d_0 = 0$
This seems to work out to pretty much the same distance as the standard formula for my specific case, with intervals of about $5 ms$, even over longer periods of total time.
But let's say I start from $0$ and have the acceleration $1 m/s^2$ during $1 s$. The standard formula gives me
$$d = 0 \cdot 1 + \frac{1 \cdot 1^2}{2} = \frac{1}{2} = 0.5 m$$
and the other formulas give me
$$v = 0 + 1 \cdot 1 = 1m/s$$
$$d = 0 + 1 \cdot 1 = 1m$$
I'm not entirely sure why they seem to produce the same results on the values in my Excel spreadsheet, but not in the example above? I have a gut feeling it's related to averages, integrals and infinitely small time intervals somehow. Basically I suspect that my $0.005 s$ intervals are small enough to somehow make the formulas almost identical...
Can someone explain in a pedagogic way what the difference is (and what the problem with the "cumulative" formulas are)?
Note that your formula
$$d = v_0 \cdot t + \frac{a \cdot t^2}{2}$$
is only valid if the acceleration is constant though that is not your problem here.
Your problem is the size of your sample intervals. Your approach is using the same velocity for the whole of each interval. With sufficiently many intervals, this can give a good approximation but the fewer the intervals, the poorer it will be. The worst case will be a single interval. Your final calculation assumes that the velocity is $1m/s$ for the whole of the first second. Assuming constant acceleration, you could improve the results by using the average velocity in each interval rather than the final.