Bricklayer problem (involves springs)

[lowellite]

Homework Statement

An ingenious bricklayer builds a device for shooting bricks up to the top of the wall where he is working. He places a brick on a vertical compressed spring with force constant k=450N/m and negligible mass. When the spring is released, the brick is propelled upward. If the brick has mass 1.80kg and is to reach a maximum height of 3.6m above the initial position on the compressed spring, what distance must the spring be compressed initially?

Homework Equations

W=Fs
W_total=K_f-K_i
F=kx (Hooke's law)

The Attempt at a Solution

Someone else has already solved this problem (but with k=350 instead of 450) https://www.physicsforums.com/showthread.php?t=254633&highlight=bricklayer", though I can't really understand much of it. They said that the total work done on the brick is 0, but I'm not really sure why. Is it because the velocity of the block is 0 when the spring begins to decompress and when the block reaches its maximum height (so the initial and final kinetic energy would both be 0)? For some reason, I can't seem to understand this on a conceptual level.

I'm also not very sure about the rest of the solution. So the work done to compress the spring is W=0.5kd², where d is how much the spring was compressed. Does that mean when the spring decompresses, an equal amount of work is done?

The work done by gravity from the time when the brick leaves the spring to the time when the brick reaches its maximum height is W=-mgh, where h is the maximum height (3.6m).

Then I'm supposed to relate W=0.5kd² and W=mgh to get 0.5kd²=mgh, but I don't really understand how this can be done...

 

[Clever-Name]

The total work is 0 because if you consider just potential energy then the potential energy stored in the spring is used to propel the brick into the air; where at the point of K=0 all of the potential energy is then gravitational.

It's easier to think of this purely as conservation of energy. There is an initial point and then a final point. Since we don't consider things like air resistance and energy lost from heat you can just look at time 0 (spring compressed) and time f (3.6m high)

 

[Ray Vickson]

At the top of its flight all the energy "in" the brick is potential. At the start of its flight, all the energy in the brick is kinetic. Before the start of its flight, all the energy that will go into the brick is potential energy stored in the spring.

RGV