Solving Friction Fraction: Step-by-Step Guide

[Mivz18]

Here is the problem:

A student has two ramps both are at an angle of 30o. Ramp 1 is frictionless and ramp 2 has friction. The student also has two blocks, one for each ramp. She pushes the blocks up the ramps with the same initial velocity. The block on ramp 2 only travels a fraction f = 0.625 as far before coming to a stop as the block on ramp 1. Find the coefficient of sliding friction between the block and ramp 2.

How do I even begin this problem? I have found the forces of each of the blocks. I know that it will involve more than one equation and the canceling out of variables to find the unknown needed. But how?

 

[NateTG]

I think I would be inclined to use work-energy calculations. Both blocks start with the same amount of energy...

 

[Mivz18]

Well, this is a problem I've put aside a while, but still haven't been able to figure out. So far, I have discovered that Vo = Vo for both blocks on ramp 1 and 2. D1 = D and D2 = 0.625D .

Solving for A), Ramp 1

Vf ^ 2 = Vi ^2 + 2a(delta D)
0 = Vi ^2 + 2a(delta D)
0 = Vi^2 + 2aD

Solving for B), Ramp 2

Vf^2 = Vi^ 2 + 2a(delta D)
0 = Vi^2 + (2a)(0.625D)

Substituting and setting equations equal you get:

Vi1 ^2 + 2aD = Vi2^2 + (2a)(0.625D)
2aD = (2a)(0.625D)

From here, if I try to solve for one variable, both cancel out leaving me at a road block. Did I take a wrong detour in coming to where I am now? Or have I done something completely wrong?

 

[Mivz18]

Nevermind, I figured out where to go and how a1 and a2 were defined to cancel out the D variable and bring in the coefficient of friction variable.