The maximum speed of a mass m on an oscillating spring is vmax . what is the speed of the mass at the instant when the kinetic and potential energy are equal?

Let
A =  the amplitude of vibration
k =  the spring constant
m =  the mass of the object

The displacement at time, t, is of the form
x(t) = A cos(ωt)
where
ω =  the circular frequency.

The velocity is
v(t) = -ωA sin(ωt)

The maximum velocity occurs when the sin function is either 1 or -1.
Therefore

Therefore


The KE (kinetic energy) is given by


The PE (potential energy) is given by


When the KE and PE are equal, then


For the oscillating spring,

Therefore