Vertical throw and free fall are particular cases of uniformly accelerated motion under the influence of gravity. Assuming a constant gravitational acceleration \(g\) (on Earth's surface, approximately \(9.8\,\text{m/s}^2\)), we can describe the position and velocity of the object as a function of time:

Position: \[ y(t) = y_0 + v_0\,t \;-\; \frac{1}{2}\, g\, t^2 \] where \(y_0\) is the initial height and \(v_0\) is the initial velocity.

Velocity: \[ v(t) = v_0 \;-\; g\,t \]

The maximum height is reached when \(v(t) = 0\). Solving for time and substituting into the position equation, we obtain:

Maximum Height: \[ h_{\text{max}} = y_0 \;+\; \frac{v_0^2}{2\, g}. \]

This simulator numerically integrates these equations to display the trajectory, velocity, and maximum height in real-time.