Track object in space and time

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I'm currently struggling to solve the following problem: The goal is determine the movement of an object by tracking an object in 2d-space and time (third dimension).

I always know the $x$ and $y$ position of said object throughout a certain time course $t$. For example:

t    x     y
1   1.1   0.4
2   1.2   0.5
3   1.4   0.5
...

Now, I'm struggling with how to determine velocity and acceleration vectors for this object in the $t$-dimension. So, I don't want to calculate these vectors given the $x,y$ information, but track this point on the t-dimension by using its velocity and acceleration vectors.

Unfortunately, this is where I'm stuck and I'd be happy about any help I can get!