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# Moving up Ramp: Friction and Energy

APPH12-8N7KD4

A box is given an initial push up a ramp that is inclined at angle $\alpha$ above the horizontal, and then allowed to slide freely up the ramp until it stops at a maximum height $H$, measured relative to the base of the ramp. The box's speed at the base of the ramp is $V$, and the coefficient of kinetic friction between the box and ramp is $\mu$.

Which one of the following equations is a correct application of work and energy ideas, that could be used to solve for the maximum height $H$ the box reaches above its starting position?

A

$\cfrac { 1 }{ 2 } m{ V }^{ 2 } + \cfrac { \mu mgH\sin { \alpha } }{ \cos { \alpha } } = mgH$

B

$\cfrac { 1 }{ 2 } m{ V }^{ 2 } - \mu mgH\cos { \alpha } = mgH$

C

$\cfrac { 1 }{ 2 } m{ V }^{ 2 } + \mu mgH\sin { \alpha } = mgH$

D

$\cfrac { 1 }{ 2 } m{ V }^{ 2 } - \cfrac { \mu mgH\cos { \alpha } }{ \sin { \alpha } } = mgH$