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An outlaw cuts loose a wagon with two boxes of gold, of total mass 300 kg, when the wagon is at rest 50 m up a 6.0$^\circ$ slope. The outlaw plans to have the wagon roll down the slope and across the level ground, and then fall into a canyon where his accomplices wait. But in a tree 40 m from the canyon's cliff wait the Lone Ranger (mass 75.0 kg) and Tonto (mass 60.0 kg). They drop vertically into the wagon as it passes beneath them $(\textbf{Fig. P8.99})$. (a) If they require 5.0 s to grab the gold and jump out, will they make it before the wagon goes over the cliff? The wagon rolls with negligible friction. (b) When the two heroes drop into the wagon, is the kinetic energy of the system of heroes plus wagon conserved? If not, does it increase o Figure P8.92 r decrease, and by how much?

A. $34.9 \mathrm{m}$B. $4.8 \times 10^{3} \mathrm{J}$

Physics 101 Mechanics

Chapter 8

Momentum, Impulse, and Collisions

Moment, Impulse, and Collisions

Cornell University

University of Michigan - Ann Arbor

Hope College

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so problem 8.99. The ah set up here is is ah, bandit of some. You has loosed a wagon containing some gold and is planning to have it roll off this cliff where his friends could then take the gold and then they get away with it or whatever and become rich. Total master, that is 300 kilograms. It starts off rolling down this shallow incline, but then 40 meters from the cliff, the lone ranger and pronto drop onto the wagon and are hoping to be able to grab the gold and get it off the wagon before it um goes over the class. However, we're told that it takes about five seconds for them to get the golden jump out. And we need to find out whether the wagon will go off the cliff within that time before they could escape. And then we're also curious of whether the potential of the kinetic energy of the wagon and the heroes is conserved. And if not, how much does it change by? So for our first part here, we want to figure out how fast the wagon is going before they jump in. And so This is a conservation of energy thing. So the initial potential energy from it being of this height it's going to be people to the kinetic energy once it's rolling on the ground with negligible resistance. So mg times now we have the height of this is gonna be 50 meters times the sight of six degrees. This is going to be equal to the kinetic energy. So 1/2 piece where and that's loving this for for V. Because see that he is going to be 10.12 meters per second out for this collision. We can think of it as a collision. Uh, we need to use the conservation of momentum like in every sort of collision problem. I know what we would expect to happen is they have zero momentum in the ex direction and so in order to, But they have. They have mass. And so, in order for the center of mass of the system consisting of two people and wagon to be moving at the same velocity, this will have to slow down. So we have now. That's, uh let's call it the wagon, eh? And both of them together be so M a times the be here is going to eat all the mass of wagon, plus the combined mass of the Lone Ranger and pronto times its final velocity or speaking now solving this for the F we have that it's equal to ratio of the initial mass of the cart to the final mass. Time is how fast it is going to begin with. Hey! And so that's, uh, 6.98 meters per second. So now 6.9 eight meters for second times, the five seconds they have to take 34. Well, maybe you make that look like 34.9 meters so they can get out off the wagon before it goes over and get See? Now we know how fast the car was going to begin with, and we know how fast it was going after they jump in. And we're to assume that they jump in from ah, very short height, relatively speaking, and so they don't really have any kinetic energy of their own. And so now we want to see what change, if any, there is in the kinetic energy. And since this is an any elastic collision, we sort of should know to begin with, said the energy. The kinetic energy will not be conserved, so we need to find how much it changes by. Actually, it's just a final. Okay, initial. So the final kinetic energy is this tape 1/2 out of everything so that not dragging it around everywhere and they plus M b times the F squared. And then it's initial kinetic energy. It's just m A that we call the Squared. And so this is negative. It's not negative. Then we're in trouble. 4.8 Killer Jules. So this is the amount by which the kinetic energy of the system as deep peace from them jumping into it and staying in it. If you were in the lasting collision, they would have to somehow bounce off it. It should be strange. So this is good. It makes sense. They they can get out of the cart with the golden time, so everything has a happy

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