Finding initial (vertical) velocity from a vector - projectile motion

Question

Only a starting point $(p)$ and initial velocity $(v)$ is given in a form of cartesian points $(p.x, p.y)$ and $(v.x, v.y)$. I want to find the equation of the parabola going though them so then I can easily find the height at the given time. Maybe my approach is wrong and I should look into different approaches?

Answer 1

If you want to find an equation that describes the current position $\vec{x}(t)$, you set up an equation of motion $$m\cdot\frac{\mathrm{d}^2 \vec{x}(t)}{\mathrm{d}t^2} = \vec{F}$$

where $\vec{F}$ is a force that you have to specify (can be gravitational force, Lorentz force, etc.). I assume, for your scenario it is likely the gravitational force applied onto your projectile. Thus

$$m\cdot\frac{\mathrm{d}^2 \vec{x}(t)}{\mathrm{d}t^2} = m\cdot \vec{g}$$

This is an ordinary differential equation of degree 2 and its solution is given by

$$\vec{x}(t) = \frac{1}{2}\vec{g}t^2 + \vec{v}_0t + \vec{x}_0$$

Where $\vec{v}_0$ and $\vec{x}_0$ describe the initial speed and the initial position, respectively.