Energy changes in mass spring system

[Asad Raza]

Why is the kinetic energy vs time curve of a mass spring system is just a modulus of respective velocity time curve.
How can we prove it mathematically ?

 

[ShayanJ]

Actually that's not true!
Step1) Consider the most general trajectory for a harmonic oscillator:$x(t)=A\cos \omega t+B\sin\omega t$
Step2) Differentiate it w.r.t. time to get the velocity $v(t)$.
Step3) Put $v(t)$ in $K(t)=\frac 1 2 m v(t)^2$.