How do I find final velocity using a graph of non-constant acceleration?

Question

I have a series of data representing the acceleration of a falling object with a parachute. While the chute opens up, in the first instance it starts of in the high -9 m/s/s range (- being down) and then cuz of the chute and the drag, it starts to lessen. The final three valuea before it hits the ground are:

5.9m/s/s

5.5m/s/s

5.1m/s/s

We used an accelerometer to measure it. How do I work the final velocity from all the data?

Answer 1

one way would be to fit a curve through the data which will give you some relation for a as a function of time:

a = f(t) = dv/dt

dv = f(t)dt

And then do the integration.

An estimate would be to use the mean accleration over each time interval and then times it by the time to get the velocity increment during that time and then sum them.

SUM { t*(a(k) + a(k-1)]/2 + t*(a(k - 1) + a(k - 2)/2 + ... + t*(a(2) - a(1)0/2 }

Which can be rewritten as

t* SUM{ a(k)/2 + a(k-1) + a(k-2) + .... + a(1)/2 }

You can see that this is good for constant a since:

v = ta*(k - 1) where k is the number of points and t is the time interval

Under constant acceleration v = aT but T = (k - 1)t

Answer 2

get the equation of the acceleration graph

then take the antiderivative

substitute for the time you're looking for and thats it

Answer 3

You just use

v= final velocity

u = intial velocity

v = 0.5at^2 +u