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Kinematic equations of SHM in terms of time

The displacement $$(\vecphy{x})$$, velocity $$(\vecphy{v})$$ and acceleration $$(\vecphy{a})$$ of an object undergoing simple harmonic motion in terms of time $$(t)$$ are given by different sets of equations.

Like all kinematics equations, these can only be applied to linear motion with respect to a particular axis with a specified positive direction. This means that the quantities are represented as scalars, with negative signs to imply motion in the opposite direction.

When the motion begins from the equilibrium position (i.e. when $$t=0$$, $$x=0$$), the displacement function is a sine. When the motion begins from an extreme position (i.e. when $$t=0$$, $$x=\pm x_{0}$$), the displacement function is a cosine.

Equilibrium position Positive extreme position Negative extreme position
$$$x=x_{0}\sin\omega t$$$ $$$x=x_{0}\cos\omega t$$$ $$$x=-x_{0}\cos\omega t$$$
$$$v=\omega x_{0}\cos\omega t$$$ $$$v=-\omega x_{0}\sin\omega t$$$ $$$v=\omega x_{0}\sin\omega t$$$
$$$a=-\omega^{2}x_{0}\sin\omega t$$$ $$$a=-\omega^{2}x_{0}\cos\omega t$$$ $$$a=\omega^{2}x_{0}\cos\omega t$$$

The difference in the equations reflects the fact that the cosine function is just a moved sine function (at $$t=0$$, the cosine function takes value 1 and the sine takes value 0).

$$x_{0}=$$maximum displacement; $$\omega=$$angular frequency.