I know this formula D = vt + 1/2at^2* for calculating the distance given initial velocity, time and acceleration. But what if my acceleration is not static, but increasing exponentially defined by f(n) = a^n? If the acceleration a starts increasing at distance x and time tx (reference point a) and I do a measurement after n seconds (reference point b) with the formula above, the results would be incorrect, because the acceleration initially was smaller.
Example: at reference point a object travels at velocity 10 m/s and acceleration is 2 m/s^2 and time is 40 seconds (tx). After this point acceleration starts to increase by a(n) = a^n where n is every second after tx. So at n=2 (total 42 seconds) seconds acceleration would be 4 m/s^2, n=3 it would be 8 m/s^2 and so on. I want to measure the distance from a to b, where b is after n=10 (total 50 seconds).
Example2: at reference point a object travels at velocity 10 m/s and acceleration is 2 m/s^2 and time is 40 seconds (tx). After this point acceleration starts to increase by a(n) = a^dn where dn is every meter after a. So at dn=2 (total a+2 meters) seconds acceleration would be 4 m/s^2, dn=3 it would be 8 m/s^2 and so on. I want to measure the distance from a to b, where b is after after 10 seconds (total 50 seconds).