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Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 $\mathrm{kg}$ , is shiding to the left at 5.00 $\mathrm{m} / \mathrm{s}$ , while the other, of mass 5.75 $\mathrm{kg}$ , is slipping to the right at 6.00 $\mathrm{m} / \mathrm{s}$ . They hold fast to each other after they collide. (a) Find the magnitude and direction

of the velocity of these free-spirited otters right after they collide. (b) How much mechanical energy dissipates during this play?

(a) $v_{2}=0.226 \mathrm{m} / \mathrm{s}$ to the left

(b) $D . M . E=197 \mathrm{J}$

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in this question. We have two orders traveling, one is traveling to the right and another one is traveling to the left. The one that is traveling to the right goes with 6 m per second and it has a mass off 5.75 kg. The one that travels with the left goes with a velocity off 5 m per second and it has a mass off 7.5 kg. In the first item off this question, we have to determine what is the velocity off. Both others after the collision notes that after the collision they will be traveling together to solve the first item. You have to remember about conservation of momentum, that losses, the following The net momentum of the situation is conserved either before and after. It's the same. So we can say that the net momentum before let me call this Pew net is because of the net momentum. After the collision, which I'm calling P prime net now, I will choose a reference frame in my reference frame. Everything that is pointing to the right is going to the positive direction. Now note that the two waters are traveling initially, but one order is traveling to the right and another one is traveling to the left, meaning that this order, which is traveling to the right, will have a positive velocity. In my reference frame on this order that is traveling to the left, we will have a negative velocity in my reference frame. Okay, Now let's calculate their momentum. The momentum is given by this equation, so we can say the following Before the collision, we had this order with a mass off 5.75 kg, traveling to the right so with a positive velocity off 6 m per second. And we also had another order that is traveling to the left. So it has a negative velocity. And then we get a minus sign here. It's traveling to the left, but it has a mass off 7.50 kg. So we have 7.50 times the velocity, which is 5 m per second. And then after the collision, both others will travel together with the same velocity which I'm calling V. Okay, so after the collision, the net momentum is given by the mass off the two orders times the velocity of the two authors so to solve the first item. All we have to do is so for V. So we is equal to 5.75 times six minus 7.50 times five divided by 5.75 plus 7. 50 on This results in a speed off approximately minus 0.226 m per second, meaning that after the collision the otters will travel to the left because of this minus sign with a velocity off 0.226 m per second. And this is the answer to the first item in the second item, we have to deter mined the amount off dissipated mechanical energy in this situation. So all we have to do is evaluate the kinetic energy before and the kinetic energy after and then compare both off these energies before the collision, we had a kinetic energy that I'm calling e k. And that energy is given by the some off the kinetic energy off this order or if the kinetic energy off this order order Okay, The kinetic energy is given by this equation. So we have the following the kinetic energy off. This order is given by its mass which is 5.75 times velocity squared. Then we add magnetic energy off the order order, which is given by its Mass 7.15 times its velocity five squared. Notice that I'm not plugging in the minus sign here because we're square in the velocity anyway. So that minus sign will make no difference in the end. Then, after the collision, they have a kinetic energy e k prime that is given by one half off the total mass 5.75 plus 7.50 times the final velocity squared. Then we have 0.226 squared so that the variation in the kinetic energy is given by the final kinetic energy one half off, 5.75 plus 7 50 times 0.226 squared minus Danish magnetic energy, which is one half off 5.75 times six squared minus because we had plus plus so minus one half times 7.50 times five squared. These results in a variation in the kinetic energy that is given by approximately minus 197 jewels, meaning that 197 jewels off energy were dissipated in this situation,

Brazilian Center for Research in Physics