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Calculus Notation

<aside> 💡 Differential calculus notation is used to represent rate of change

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Velocity

<aside> 📕 Velocity is defined as the rate of change of displacement.

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$$ v = \frac{ds}{dt} $$

where:

• $v$ is velocity, in $ms^{-1}$.
• $s$ is displacement, in $m$.
• $t$ is time, in $s$.

Acceleration

<aside> 📕 Acceleration is defined as the rate of change of velocity.

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This also means that it is the second derivative of displacement.

$$ a = \frac{dv}{dt} = \frac{d^2s}{dt^2} $$

where:

• $a$ is acceleration, in $ms^{-2}$.
• $v$ is velocity, in $ms^{-1}$.
• $s$ is displacement, in $m$.
• $t$ is time, in $s$.

Derivations

1st Equation of Motion

Start with definition of acceleration as the rate of change of velocity.

$$ a = \frac{dv}{dt} $$

$$ a\int_{0}^{t} dt = \int_{u}^{v} dv $$

$$ a[t]_0^t = [v]_u^v $$

$$ at - 0 = v - u $$