Acceleration function of position and time

Question

I have an acceleration function in python with position and time parameters and returns the acceleration value.

I need the end velocity at a position ,start velocity is zero.

how to calculate this problem?

Answer 1

Velocity $\vec v(\vec r,t)$ is the integral of acceleration $\vec a(\vec r, t)$ with respect to time, and position $\vec r(t)$ is the integral of velocity with respect to time. So if the object starts at the origin with velocity zero then at time $t$

$\vec v(\vec r(t) ,t) = \int_0^t \vec a(\vec r(t), t) dt \\ \vec r(t) = \int_0^t \vec v(\vec r(t), t) dt$

Unless your acceleration function is something simple, the best you can do here is numerical integration. The simplest method of numerical integration is Euler's method. With this method, you take a small time step $\delta t$ and assume that velocity and acceleration are constant over that time step. Then you update position and velocity for the beginning of the next time step as follows:

$\vec r(t + \delta t) = \vec r(t) + \vec v(\vec r(t), t) \delta t \\\vec v(t + \delta t) = \vec v(t) + \vec a(\vec r(t), t) \delta t$

There are other methods of numerical integration which are more complicated, but give more accurate results.