Find Next Position and Velocity from Instantaneous Values

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To find the position of an object at a given point in time: $y_0 + v_0t - \frac{32t^2}{2} = y_t$.

And to find the object's speed at a given point in time: $v_0 - 32t = v_t$

So say I give the following values:

  • $y_0 = 500$
  • $v_0 = 2$

Here is the plot of $y_t$:$v_t$ as t changes:

  1. 486:-30
  2. 440:-62
  3. 362:-94
  4. 252:-126
  5. 110:-158
  6. -64:-190

If I'm only given a point and velocity on this curve and I'm not told what t is at that point; is it possible to find the next position and velocity?

So for example, say I'm told:

  • $y_{i-1} = 362$
  • $v_{i-1} = -94$

Now I need to find $y_i$ and $v_i$.