Using a simply pulley/rope system, a crewman on an Arctic expedition is trying t

Question

Using a simply pulley/rope system, a crewman on an Arctic expedition is trying to lower a 6.43-kg crate to the bottom of a steep ravine of height 22.9 meters. The 55.6-kg crewman is walking along holding the rope, being careful to lower the crate at a constant speed of 1.50 m/s. Unfortunately, when the crate reaches a point 12.7 meters above the ground, the crewman steps on a slick patch of ice and slips. The crate immediately accelerates toward the ground, dragging the hapless crewman across the ice and toward the edge of the cliff. If we assume the ice is perfectly slick (that is, no friction between the crewman and the ice once he slips and falls down), at what speed will the crate hit the ground? Assume also that the rope is long enough to allow the crate to hit the ground before the crewman slides over the side of the cliff. At what speed will the crewman hit the bottom of the ravine? (Assume no air friction.)

Explanation / Answer

a)

We will use Conservation of energy to answer this part :

KE of the system at the start of the acceleration :

0.5*(55.6 + 6.43)*1.5^2 = 69.8 J

Now, the PE of the crate = 6.43*9.8*12.7 = 800 J

Now, at the lowest point of the crate, all the PE converted to KE of the system:

So, 800 = 0.5*(6.43+55.6)*(v^2) - 69.8

So, v = 5.3 m/s <-----answer

b)

Now for the crewman :

All its PE at top is converted to KE at bottom

So, 55.6*9.8*22.9 = 0.5*(55.6)*(v^2 - 5.3^2)

So, v = 21.8 m/s

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