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# Kinematic equations of SHM in terms of displacement

The equations for the velocity $(\vecphy{v})$ and acceleration $(\vecphy{a})$ in terms of displacement $(\vecphy{x})$ can be derived from the equations in terms of equilibrium or extreme positions and the equation for displacement in terms of time ($x=x_{0}\sin\omega t$).

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.

\begin{align*}v&=\pm\omega\sqrt{x_{0}^{2}-x^{2}}\\a&=-\omega^{2}x\end{align*}

The equation for acceleration is the defining equation for simple harmonic motion (showing the proportionality of $a$ and $x$).

Some remarks:

• $x$ is and must always be smaller than $x_{0}$ for the velocity to be a real number.

• The $\pm$ sign corresponds to the fact that the harmonic oscillator periodically changes direction, with velocities of the same magnitude but opposite directions occurring at the same displacement.

• The minus sign in $a=-\omega^{2}x$ stands for the fact that the oscillator strives to restore the equilibrium whenever the displacement is non-zero.

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