Change in Energy - Spring Constant - Moving Body - Transfer of Energy?

Question

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HOW DO U DO THIS I CANT GET TEH ANSWER PLZZ HELP!

Answer 1

Calculate the total energy of the package when it's 25 feet from the spring
Total Energy = Kinetic Energy + Gravitational Potential Energy
You need mass for kinetic energy: m = Wt/g. You need to do some trig to figure the h for potential energy (pe = Wt*h). You can leave the units for both types of energy in lbs*ft.

By the time the package reaches the spring, all the potential energy will have converted into additional kinetic energy (you neglect friction). So the total energy will be the same as it was 25 feet up the slope. The spring will have to do an amount of work equal to that total energy.

Wk = Fave*d
Since the spring is compressed to begin with you need to calculate Fave as
Fave = (Fmin + Fmax) / 2
Fmin = (125 lb/in)*6 in
Fmax = (125 lb/in)*(d+6 in)
So
Fave = ((125 lb/in)*6 in + (125 lb/in)*(d+6 in)) / 2
Fave = ((125 lb/in)*12 in) + (125 lb/in)*d) / 2

And the work the spring does is
Wk = Fave*d
Wk = ((125 lb/in)*12 in + (125 lb/in)*d)*d / 2

Set the total energy from above equal to this Wk expression. Looks like you will have a quadritic equation. Probably not fun, but doable.