A truck with mass `m` has a brake failure while going down an icy mountain road of constant down and slope angle a (see figure). Initially, the truck is moving downhill at speed `V`. After careening downhill a distance `L` with negligible friction, the truck driver steers the runaway vehicle onto a runway truck ramp of constant upward slope angle `beta` . The truck ramp has to soft sand surface for which the coefficient of rolling friction is `mu_(r)` What is the distance that the truck moves up the ramp before coming to a halt?

Options

A. `((nu_(0)^(2)//2g)+_Lsin alpha)/(sinbeta-mu_(r)cosbeta)`
B. `((nu_(0)^(2)//g)-L sin alpha)/(sinbeta+mu_(r)cosbeta)`
C. `((nu_(0)^(2)//2g)+Lsinalpha)/(sinbeta-mu_(r)cosbeta)`
D. None of these

Answer

a

Explanation

Donote the distance the truck moves up the ramp by `x. K_(1)=1/2mnu_(0)^(2),U_(1)mgL sin alpha, K_(2)=0,U_(2)=mgx sin beta and W_(other)=mu_(r) mgx cos beta`. From `W_(other)=(K_(2)+U_(2))-(K_(1)+U_(1)),and solve for x`,
`x=(K_(1)+mgLsinalpha)/(mg(sinbeta+mu_(r)cosbeta))=((nu_(0)^(2)//2g)+Lsinalpha)/(sinbeta+mu_(r)cosbeta).`