Solve this Physics Problem

Question

A girl on a skateboard (total mass of 40 kg) is moving at a speed of 13 m/s at the bottom of a long ramp. The ramp is inclined at 11° with respect to the horizontal. If she travels 14.0 m upward along the ramp before stopping, what is the net frictional force (in N) on her? (Assume the positive direction is upward along the ramp. Indicate the direction with the sign of your answer.)

Answer 1

It can be shown (see comments below) that

v₀² = 2gx[sin(θ) +μ_k cos(θ)]

(v₀²/2gx) – sin(θ) = μ_k cos(θ)

μ_k = (v₀²/2gxcos(θ)) – tan(θ)

μ_k = (169/2(9.8)(14)cos(11°)) – tan(11°) ≈ 0.433

Answer 2

We can use conservation of energy.

If no friction were there, then energy would be conserved: at first, there is only kinetic energy (K = 1/2 mv^2); when the girl stops, there is only potential energy (U = mgh = mgLsin(theta)), and the two of them would be equal [L0 = length along the ramp with no friction, theta = ramp angle].

Because of friction, some energy is lost into work of the friction force, i.e.

K - U = frictional force * L

0.5 * m * v² - mg L * sinθ = force * L

L=14m. This can be solved to find the friction "force", which clearly is negative, if the positive direction is upward along the ramp.