Simple Harmonic Motion Question [closed]

Question

I have two formulae:

Displacement = Amplitude * Cos(Angular Frequency * Time)

Velocity = - Amplitude * Angular Frequency * Sin(Angular Frequency * Time)

OR

$x = Acos(wt)$

$v = -A.w.sin(wt)$

And the question is using these two formulae show $v = \pm W\sqrt{A^2 - x^2}$

Thanks for your help in advance!

Answer 1

Calculate $x^2 +\frac{v^2}{\omega ^2}$:

$x^2 + \frac{v^2}{\omega ^2}=A^2(cos^2(\omega t) + sin^2(\omega t) )$

$x^2+ \frac{v^2}{\omega ^2}=A^2$

$v^2=\omega ^2(-x^2 + A^2)$

Which gives us

$v=\pm \omega \sqrt{-x^2+A^2}$