Velocity from acceleration equation

Question

I've been working with an accelerometer, and was assigned the job of trying to find the velocity from the accelerometer, I take acceleration data every 10 ms (sampling rate 100KHz) and I used the equation : Vf = V0 + a*t, where t is 10 ms

V0 is 0 at first, and then it takes the old Vf value

I tried moving the accelerometer in Y+ and then Y- direction, using only the y-axis.

And it works okay, see the figure below :

but I recently saw that this equation is designed to be used in an application with a constant acceleration, how come it works for me? Or am I wrong here?

Answer 1

Let's look at where that equation comes from: $$ a_\mathrm{ave} = \frac{\Delta v}{\Delta t} $$

The average acceleration during a time interval is how much the velocity changed, $\Delta v = v_f - v_i$ divided by the length of the time interval $\Delta t$. If $a$ is constant than $a_\mathrm{ave}$ is just $a$.

We can solve that equation for $v_f$ and get your equation: $$ a_\mathrm{ave} = \frac{v_f - v_i}{\Delta t} \implies v_f = v_i + a_\mathrm{ave} \Delta t$$

The key here is that for $a$ to equal $a_\mathrm{ave}$, it doesn't need to be constant over the whole time of the experiment. It only needs to be constant over each $10$ ms interval. If the time interval is short enough, then $a$ will be approximately constant, and the equation is a good model for what's going on.

In practice your accelerometer likely determines $a$ by averaging over the $10$ ms time interval between outputs. So it's probably telling you the average acceleration in each interval anyway.