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I'm really confused how we have a distance formula from acceleration. I understand acceleration is the change of velocity/time, however I don't understand how you can calculate a distance based on a velocity that is changing with every instant of time. I understand that you multiply time squared against the rate of the acceleration and dividing in half but I am still having a hard time wrapping my head around why this works. If the speed is constantly changing and there is no amount of time that the speed stays at one speed how is it possible to calculate distance?

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Integrating acceleration gives velocity as a function of time. Integrating that equation gives position as a function of time (assuming acceleration is constant).

This is all derived from a = dv/dt

How this works, given an acceleration, a starting speed, and a time, is by segmenting the problem into many tiny pieces and summing them together. This occurs in the integration process.

At each "piece" in the timeline or a time step of whatever is traveling, we can find how fast it is going at that instance (because we have constant acceleration and v= at. This process is not perfect since there are infinite half steps between any given value, but it works very well for close approximations.

For the instance or "piece" that we know the speed for, we are able to figure out how far the object in question has traveled (because we have the time and speed). Since the acceleration is constant, we can figure out what the speed would be for every next time step along the way. Since we know the speed for each time step, we can find the distance traveled at each time step.

We do this until we reach the desired time and then add up each distance for all the time steps and boom! We have our position.

The kinematics equations you are given to solve these equations have already been figured out to make life easy on you.

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