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- TL;DR Summary
- Using only the current component-wise velocity and position, the trajectory of a projectile needs to be found. The object may change velocity mid flight, an the trajectory should reflect any changes.

Is it possible to trace the trajectory of an object using only its velocity and position, both of which are given as components. My method of doing so involves using the time until max height is reached, and using that time value to calculate the max height itself (h,k), then plugging in the current point to find the constant a in the equation y=a(x-h^2) +k. I used the following equations, but noticed that as the vertical velocity decreased due to gravity, the max height also changes which it doesn't in reality. I am now wondering if what I'm trying to do is possible or not.

$$time_{max} = \frac{v_{y}}{g}\space \space(1)$$

$$x_{max} = x_{current} + v_{x} * time_{max}\space \space(2)$$

$$y_{max} = y_{current} + \frac{{v_{y}}^2}{2a}\space \space(3)$$

$$parabola: a(x-x_{max})^2 + y_{max}\space \space(4)$$

$$time_{max} = \frac{v_{y}}{g}\space \space(1)$$

$$x_{max} = x_{current} + v_{x} * time_{max}\space \space(2)$$

$$y_{max} = y_{current} + \frac{{v_{y}}^2}{2a}\space \space(3)$$

$$parabola: a(x-x_{max})^2 + y_{max}\space \space(4)$$