Stella Academy

mathematics Module

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Lesson Directive // Derivatives & Rate of ChangeREF_CORE

Instantaneous Change

v(t)

=

d

dt

s(t)

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

v(t)

Velocity

m/s

s(t)

Position Function

t

Time

While algebra gives average speed over a long trip, calculus allows us to find the exact velocity of a spacecraft at a single instant in time. This is the foundation for solving complex Differential Equations.

INSIGHT: A derivative is simply a rate of change at a specific moment.

The Derivative

Taking the derivative of a position function gives the velocity function. Taking the derivative of velocity gives acceleration. These precise calculations allow us to model complex Trajectories.

INSIGHT: Position -> Velocity -> Acceleration.

Rocket Trajectories

As a rocket burns fuel, its mass changes constantly, and its acceleration increases. Calculus is essential to model these dynamic, continuously changing systems.

INSIGHT: Calculus helps predict complex, changing motion.

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Syllabus Worked Example

Q: A probe's distance from a moon is given by s(t) = 5t^2 + 10t meters. Find its velocity v(t) at t = 3 seconds.

1.Take the derivative of position: v(t) = \frac{d}{dt}(5t^2 + 10t).

2.Apply the power rule: v(t) = 10t + 10.

3.Substitute t = 3: v(3) = 10(3) + 10.

4.Calculate the result: 30 + 10 = 40.

Result:v(3) = 40 \text{ m/s}