Silly Question about Potential Energy

[nhmllr]

I gave myself a fun little problem to pass the time
I have this clicky pen, that retracts and pops out the point when you press the button on the back
If you press the button down on a table and let go, the button makes the pen pop up a few inches
I was curious- if you knew the maximum height the pen popped up (h_m)
the mass of the pen (m)
and acceleration due to gravity (g)
what is the energy created by the button?

My first method was thinking of the height as function of time (a parabola) of equation
h = -g/2 * t² + v_i*t
The highest point/axis of symmetry is -b/2a =
-v_i/[-g*2/2] = v_ig

Then, sub that into get the max height
-g/2 * v_i²/g² + v_i (v_i/g)=
-v_i²/2g +v_i²/g =
v_i²/2g

So, h_m = v_i²/2g
h_m * 2g = v_i²
Then KE = 1/2 * m * v²
KE = 1/2 * m * h_m * 2g
KE = m * h_m * g
Which I remember is just the equation for potential energy!
There's a connection here-- and I think I kinda of get it, but can somebody else explain?

 

[AlephZero]

When you press the button, the potential energy stored in the spring (or whatever) is converted into kinetic energy of the pen.

At the top of its flight, the velocity is zero so the kinetic energy is zero, and the energy has been converted into gravitational potentiial energy.

Assuming no energy is converted to anything else (no air resistance, no sound from the "click", etc),
KE = GPE
mv^2/2 = mgh

 

[ashishsinghal]

I agree with AlephZero