physics question: mass going up ramp with friction?

> "...what in the world is the answer!?..."

Well, this is a confusingly worded question. It makes it sound like you are STEADILY pushing the mass at 5m/sec; but if that's the case, then there is not enough information to solve the problem. (To put it differently: In that case, the distance would be "5m/sec × t", where "t" is the amount of time you pushed, which is NOT GIVEN and can't be determined. And in that case, the ramp's inclination and the coefficient of friction are irrelevant anyway.)

So, let's try a different interpretation. Maybe what they really mean is: You give the mass an INITIAL push, enough to get it going at 5m/sec, and then you LET GO of it, and see how far its inertia carries it up the ramp. That one we can solve.

I would use the work/energy theorem, which says: the change in an object's kinetic energy equals the sum of the work done on the object by all forces. Since we know the change in KE (it goes from ½mv² to zero); and since the "work" formula involves distances, there's a good chance that we can use this relationship to figure the distance.

Change in KE:

= (final KE) − (initial KE)

= 0 − ½mv²

= −½mv²

Work done by all forces:

To calculate the work, you have to consider just the components of the forces which are parallel to the motion.

GRAVITY does negative work on the mass (negative because its direction opposes the mass's motion). The component of the gravity force parallel to the slope is: "(mg)sinθ" (where θ=30°). So the work done by gravity is:

W_grav = −(mg)sinθ × d

(where "d" is the distance the mass moves along the ramp).

FRICTION also does negative work on the mass. The force of friction is "Nμ", where "N" is the normal force of the mass against the ramp; and "μ" is the coefficient of friction. It can be shown that N=(mg)cosθ. Therefore:

W_friction = −Nμ × d

= −(mg)cosθ × μ × d

By the work/energy theorem:

−½mv² = W_grav + W_friction

= −(mg)sinθ × d − μ(mg)cosθ × d

You can simplify that to:

½v² = gd(sinθ + μcosθ)

Then just solve for "d".