What is the ratio of maximum acceleration to the maximum velocity of a simple harmonic oscillator?

Hint: The acceleration is maximum at the extreme position and the velocity is maximum at the mean position.
Step 1: Find the maximum velocity.
Let x = Asinωt is the displacement function of SHM.
Velocity,                              
v=dxdt=Aωcosωt
vmax=Aω|cosωt|max
=Aω×1=ωA[|cosωt|max=1]...(i)
Acceleration,
a=dvdt=-ωA·ωsinωt
=-ω2Asinωt
|amax|=|(-ω2A)(+1)|[(sinωt)max=1]
|amax|=ω2A...(ii)                       
From Eqs. (I) and (ii), we get,
                               vmaxamax=ωAω2A=1ω
amaxvmax=ω