Who knew finding Velocity could take so much Work and Energy?

Hey everyone, first post in here, I’ll try to keep it as appropriate as I can.
To put it simply, I’m stumped. The vague setup is a vertical ideal spring attempting to push up a mass against gravity. We have everything we need to get the force, but… I can’t calculate the direct transference from acceleration into velocity. Let’s see if i can show you what i mean.
1. The problem statement, all variables and given/known data
A vertical, ideal spring whose spring constant is 875 N/m is attached to a table and compressed down by 0.160m.
Part a) What upward speed (velocity) can it give to a .380 kg ball when released?
Part b) How high above its original position (in the compressed state) will the ball fly?
2. Relevant equations
V[Velocity]= d[Distance, in our case meters] / t [time, in our case seconds]
V= a[acceleration] * t [time] (note: this is specifically for this problem, as there is no initial velocity)
Remember Gravity acting as a force acting against the upward forces (9.8m/s^2)
Force[spring] = K[spring constant]*d[distance from equilibrium, in our case the compression]
We recently were working with Kinetic and Potential energy so i’ll include these equations as well…
W[Work] = F[Force]*D[Displacement]
E<k> [Kinetic Energy]= 1/2 M [Mass]* V [Velocity]^2
W<total> [Total Work] = E<k> + E<p> (check me on this in particular)
E<p> [Potential Energy, Specifically for a Spring] = 1/2 K (spring constant) * d [spring displaced due to compression/stretch] ^2 (Credit to Doc Al for catching me, quick search through notes and google and we get this equation, which kinda stands to reason, based off our E<k> equation)
3. The attempt at a solution
I’m frustrated to no end by this particular application. I’m able to find the force applied upwards by the spring, including the counterplay by gravity and mass, but whenever i attempt to find velocity, i can only pull out acceleration- and without a time that the force is applied over, I can’t find that velocity. I suppose it could be complicated and messy if i wanted to take the distance compressed, plug it into a few formulas that would give me the time it would take to pass that spot (after passing equilibrium it would stop applying force) but that would require me to incorporate a shifting application of force as well- after all, the spring force is dependent on the distance compressed- the closer to equilibrium it gets, the less actual force it would apply. I feel like this is a problem that I’m simply approaching in the wrong way. Help me, scientificatious people of the internets! T~T


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