Negative Specific Energy Is Bound - Rotational Dynamics and Orbital Mechanics Step by Step

The specific orbital energy epsilon = v^2/2 - mu/r sums kinetic and potential terms per unit mass, and a negative value means the body is bound, lacking the energy to climb out of the gravity well. The sign verdict is exact only for the stated speed, radius, and gravity parameter in the ideal two-body model. Real orbits add eccentricity, atmosphere/drag, perturbations, central-mass uncertainty, finite-body effects, and measurement uncertainty.

Circular speed at radius four gives negative specific energy. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Circular speed-square gives two joules per kilogram

The circular speed-square 4 gives specific kinetic energy 2.

\frac{v^{2}}{2}=\frac{4}{2}=2\ \mathrm{J/kg}

Potential wins, so the specific energy is negative

Potential is negative 4 and total specific energy is negative 2.

2+-4=-2\ \mathrm{J/kg}