Rates of Change
Hover over a variable in the formula above, or see glossary below:
If Calculus is the study of change, Differential Equations are the formulas that use that change to predict the future. The expression dv/dt represents acceleration—the exact rate at which velocity changes from moment to moment.
Quadratic Drag
When falling into an atmosphere, air resistance is brutal. It's proportional to the square of velocity. If you double your speed, the drag force doesn't double—it quadruples! This creates a massive braking force.
Decay Curve
Because the deceleration depends on velocity, as the spacecraft slows down, the drag force also decreases. This creates a smooth exponential decay curve, bringing the capsule to a safe, steady speed rather than a sudden stop.
Ordinary Differential Equations (ODEs)
A differential equation is a mathematical equation that relates one or more unknown functions and their derivatives. In physics, the function usually represents a physical quantity, the derivatives represent their rates of change, and the equation defines a physical law linking the two.
The Drag Equation ODE
When a spacecraft enters an atmosphere, it experiences aerodynamic drag. The magnitude of the drag force is modelled by the equation:
According to Newton's Second Law (), and knowing that acceleration is the derivative of velocity (), we can express the deceleration of the spacecraft as a non-linear first-order Ordinary Differential Equation:
Analytical vs. Numerical Solutions
If atmospheric density () were constant, this ODE could be solved analytically using the separation of variables technique. However, in reality, atmospheric density changes exponentially with altitude, and gravity continues to accelerate the capsule downwards.
This results in a coupled system of ODEs that has no closed-form analytical solution. Aerospace engineers must use numerical integration computers to simulate the reentry trajectory, dynamically calculating the updated velocity and position frame-by-frame until touchdown.
Reference: Boyce, W. E., & DiPrima, R. C. (2012). Elementary Differential Equations and Boundary Value Problems (10th ed.). Wiley.
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