A spring has a spring constant k of 88.0 N/m. How much must this spring be compressed to store 45.0 J of potential energy

Ok

so here it is

the energ function is a square of compression

at 1 m compression you'll have a 88N force that pushes back

you might think that 1m*88N = 88J potential energy there but it's wrong

why?

because while an 88N force will start reverting the spring back to its original size when the spring comes to ... let's say 50 cm compresion that force will only be 44N and dropping to 0 till the spring gets back to full size

therefore you should think this as a geometry problem

if you draw the graph of the function F=kd

you'll have the energy as the area of the triangle defined at some d=x compression and the graph line of F=kd

the corners of the triangle would be (0,0), (x, 0), (x, kx)

therefore the area of the triangle is b*h/2

wich is x*kx=E

therefore x^2 * 88N/m = 45J

x^2 = 0.5113636m^2

x=0.715m