Stella Academy

mathematics Module

Interactive Simulation Screen

SYS_OK

Lesson Directive // Dynamic SystemsREF_CORE

Rates of Change

d

v

d

t

=

-

k

\cdot

v

^2

Hover over a variable in the formula above, or see glossary below:

v

Velocity

m/s

t

Time

k

Drag Constant

1/m

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.

INSIGHT: Differential equations link the current state of a system to how fast it is changing.

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.

INSIGHT: The v² term means a fast-moving re-entry vehicle experiences extreme deceleration initially.

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.

INSIGHT: The negative sign ensures that drag opposes the direction of motion.

●Telemetry Control Console

CAL_SYS

Adjust Simulator Parameters

Syllabus Worked Example

Q: Calculate the instantaneous deceleration (\frac{dv}{dt}) of a capsule re-entering at v = 5000 m/s, where the drag constant k = 2 \times 10^{-6}.

1.Square the velocity: 5000^2 = 25,000,000.

2.Multiply by the drag constant: (2 \times 10^{-6}) \times 25,000,000.

3.Apply the negative sign to denote deceleration: -50.

Result:\frac{dv}{dt} = -50 \text{ m/s}^2