Fly a probe through a real gravity field. Set your starting speed and direction, hit Launch, and watch the trajectory obey Newton's law (a = −GM/r², integrated semi-implicitly). Get a circle at circular velocity, an ellipse at anything less than escape, a parabola exactly at escape, a hyperbola beyond — then try the Jupiter close-encounter, near-escape and Moon-escape presets, and crank the time-warp slider to watch whole orbits in seconds.
Gravity is Newton's inverse-square law with the real G·M of each body, integrated with a semi-implicit (symplectic) Euler so energies stay accurate over many orbits. The readouts come from the vis-viva equation v² = GM(2/r − 1/a): specific energy ε = v²/2 − GM/r tells you the orbit type (ε<0 ellipse, ε=0 parabola, ε>0 hyperbola), and a = −GM/(2ε) gives the semi-major axis. Escape velocity at a given radius is v_e = √(2GM/r) = √2 × v_circular — that factor of √2 is why the curve switches from ellipse to hyperbola right around 1.414× circular. A Hohmann transfer between planets is exactly two of these ellipses joined end-to-end — burn prograde at one end to raise the other end to the destination, then burn again there to circularise. The close-encounter preset isolates the swing past a giant planet: the probe speeds up diving in and would slow back down in Jupiter's frame — but in the real solar system it keeps that extra speed because the moving planet drags it along like a sling, stealing a little of Jupiter's momentum around the Sun.