The drawing shows two boxes resting on frictionlessramps. One box is relatively

Question

The drawing shows two boxes resting on frictionlessramps. One box is relatively light and sits on a steepramp. The other box is heavier and rests on a ramp that isless steep. The boxes are released from rest at A and allowedto slide down the ramps. Which box, if either, has (a)the greater speed and (b) the greater kinetic energy at B? Providea reason for each answer. Part II The two boxes have masses of 11 and 44 kg. If A and Bare 4.5 and 1.5m, respectively, above the ground, determine thespeed of (a) lighter box and (b) the heavier box when each reachesB. (c) What is the ratio of the kinetic energy of the heavierbox to that of the lighter box at B? Be sure that your answers areconsistent with your answers to Part I. THANKS!! The two boxes have masses of 11 and 44 kg. If A and Bare 4.5 and 1.5m, respectively, above the ground, determine thespeed of (a) lighter box and (b) the heavier box when each reachesB. (c) What is the ratio of the kinetic energy of the heavierbox to that of the lighter box at B? Be sure that your answers areconsistent with your answers to Part I. THANKS!!

Explanation / Answer

(I)    since there is no friction, there are onlytwo forces acting on each box, the gravitational force (its weight)and the normal    force    the normal force, being perpendicular to thedisplacement of a box, does no work    the gravitational force does work    however, it is a conservative force, so theconservation of mechanical energy applies to the motion of eachbox    this principle states that the final speeddepends only on the initial speed, and the initial and finalheights    the final speed does not depend on the massof the box or on the steepness of the slope    thus, both boxes reach B with the samespeed    the kinetic energy is given by theexpression    both boxes have the same speed vwhen they reach B    however, the heavier box has the greatermass    therefore, the heavier box has the greaterkinetic energy (II)    (a)    according to the conservation of mechanicalenergy we get that    (1 / 2) m vB2 + m ghB = (1 / 2) m VA2 + m ghA    the speed at B gives that    vB =[vA2 + 2 g(hA -hB)]         = ........ m /s (b)    the speed of heavier box is same as thelighter box (c)    the ratio of the kinetic energies is    (KEB)heavierbox / (KEB)lighter box =mheavier box / mlighter box                                                         =..........                                                         =..........

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