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Acceleration changing over time

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Hey, i am having trouble doing the calculation on a dynamic system.

My acceleration is not constant during, and because of that my speed of the particle is changing from the original, and i wanna know how long my particle has moved in a short periode of time Δt and what the velocity is. with the math i have learned so far, i end up with only taking into account the end velocity or the beginning velocity, instead of calculating it is changing


Know is
Beginning condition: (distance, speed)
Travel: (acceleration at a specific time, that changes)
I wanna know end distance and end velocity.

At the end i need to rewrite it to a transfer function, so i have to do linearization if that can help in the calculation :)

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