A capsule entering an exponential atmosphere rho = rho0 exp(-h/H) (H = 7.2 km) obeys dv/dt = -rho v^2/(2 beta) + g sin(gamma) and v dgamma/dt = -(L/D) rho v^2/(2 beta) + (g - v^2/r) cos(gamma), where beta = m/(Cd A) is the ballistic coefficient and gamma is the path angle below the horizon. Lift (L/D > 0) flattens the path, lowers the peak deceleration and lengthens the flight.

For a straight ballistic entry (Allen-Eggers) the peak deceleration is a_max = v^2 sin(gamma) / (2 e H), reached where rho = beta sin(gamma)/H; it does not depend on beta. The heat flux uses the Sutton-Graves relation q = 1.74e-4 sqrt(rho/Rn) v^3 W/m2 with a nose radius of 1 m, and the heat load is its time integral.

Simplifications: planar flight, non-rotating Earth, fixed L/D, no ablation or radiative heating, no real-gas effects. A path that is too shallow skips out of the atmosphere; one that is too steep gives extreme deceleration. Educational only: this is not a mission design tool.