General Equation about Energy lost due to Friction

[Quantumm]

Homework Statement

The statement below arises from a marble and track lab, and I'm enthralled to figure out a generalized equation ( variables only ) for energy lost per meter of track. Track is 12 feet long, but can be curved for hills and loops.
Using a small section of track and marble, determine the average energy lost per meter of track.
The track has 2 loops and 2 hills, and starts at a certain height with potential energy only.

Homework Equations

PEi + KEi + WEi(friction) = KEf + PEf + WEf

The Attempt at a Solution

1. KE = PE - WE

2. KE = PE - μ*m*g*x(distance)

3. 1/2mv^2 = mgh - ( μ * m * g * x)

4. v^2 = 2(g * initial height) - (μ * g * x)

5. v = √(2*g*h - (μ * g * x))

6. The answer above is final velocity and you can plug that into 1/2mv^2 and compare the energy to the amount it would be without friction v = √(2gh) and find the difference between the two, which in my theory would give the average amount of energy lost per meter.

7. I'm just confused if this is the right solution or even the right direction to go with this problem.

 

[guitarphysics]

Well, the situation that you're analyzing isn't exactly general (it's a specific case of a body starting from rest at a certain height and then falling down a ramp). However, simply from the definition of work you should be able to find the energy lost due to friction. I'm assuming you don't know calc, so we'll just use basic definitions here:
W=F•∆x=µmg∆x. That's it. You basically knew it already, though :), but just hadn't realized. The energy lost due to friction is just that! If you plug formula #5 into the kinetic energy term and subtract the regular kinetic energy term with no friction, you'll find a result that agrees, which is still pretty beautiful. Love your initiative and the fact that you tried to go deeper into this, that's the best way to learn physics (in my opinion)- look at the equations yourself, play around with them a little, and try to derive some results yourself. Hope this helped :)