A truck with mass m has a brake failure while going down an icy mountain road of

Question

A truck with mass m has a brake failure while going down an icy mountain road of constant downward slope angle ? (Figure 1) . Initially the truck is moving downhill at speed v0. After careening downhill a distance L with negligible friction, the truck driver steers the runaway vehicle onto a runaway truck ramp of constant upward slope angle ?. The truck ramp has a soft sand surface for which the coefficient of rolling friction is ?r.

What is the distance that the truck moves up the ramp before coming to a halt? Solve using energy methods.

Express your answer in terms of m, ?, v0, L, g, ? and ?r.

P.S. I got this answer, but it says that it is wrong

(((v0^2)/2g)+Lsin?)/(sin? + ?rcos? )

Explanation / Answer

So:

initial energy + energy down = energy up + energy absorbed by ramp friction

init energy = 1/2 m v_0^2

energy gained going down hill = m g L sin(a)

energy up = m g X sin(b)

energy absorbed by ramp = m g mu_r X

So

1/2*m*v_0^2 + m*g*L*sin(a) = m g X sin(b) + m*g*X*mu_r

Colecting terms and simplifying:
X=1/2*v_0^2 + g*L*sin(a) / (g*sin(b) +gmu_r)

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