Hey everyone, let's check this out here. So we have a 15 kilogram box that is sliding across a frictionless surface, and its speed is going to increase. We're told that the initial speed of this block is 10 meters per second, but it's accelerating over some distance, and then at some point over here, its final velocity is going to be 30. We're told that the acceleration along this interval here, this "a", is equal to 2. That's basically all we know, and in the first problem, we want to calculate what is the change in mechanical energy of the block's mechanical energy. So the variable that we are looking for is Δme. That's the change in mechanical energy. Now remember, for a block's mechanical energy, it could be potential or kinetic. So really what they're asking for here is what is the total change in the potential energy plus the change in the mechanical energy. Alright. So let's consider each one of these terms here. Now we have a block that's not attached to any springs; there are no springs in this problem, and there's also no gravity because we're just going along the horizontal surface. So there's no change in the potential energy. So really all that happens here is that the block's kinetic energy is changing across this interval, and because it's going faster right from 10 to 30, there's a change in the mechanical energy. So that's Δme. This is just going to equal kfinal−kinitial. So Δme here is just going to equal the formula for kinetic energy is 12mvfinal²−12mvinitial². Now I have all of those values. I have the masses and the velocities, so I can actually figure this out. My Δme is just going to be 12×15×30²−12×15×10². When you go ahead and plug this all into your calculator, you're going to get 6,000 joules. So one way you can also think about this here is that there's a force that is pushing this thing, this pushing this block that's why it's causing it to accelerate. We don't know what that force is, but that force is doing work on the object, and that work is equal to the change in the kinetic energy. So that work that you're inputting changes the mechanical energy by 6,000 joules. Alright? And that's one way you can think about it as well. Let's move on to the second problem. And the second problem we want to figure out, the rate at which energy is transferred. So that's a dead giveaway that this is actually talking about power. Remember that power is equal to the change in energy over the change in time. And because we're not told because we're basically told 2 points, right, from initial to final, we don't know what's happening in between; this is an average power. So all we really have to do here is realize that the change in energy is actually just the change in the mechanical energy that we just calculated. So if we want to figure out the power, all we have to do is just have the mechanical energy, which we already have, but we also need to know the time. We don't have anything and any information about time, so that's what we need. Actually, I'm sorry. This should be a question mark. We don't have anything about time, so we're going to have to go solve that in order to get power. So let's go ahead and do that. Right? So if we want time here, let's see. We're also told that the mass of this block, the information we're told is also the initial and velocity, the acceleration, and the final velocity. These are kinematics variables. So if I want to figure out t, which is also a kinematics variable, I just need to do exactly what I did when we talked about 1-dimensional motion. I need to set up my 5 variables and I need 3 out of 5. So I don't know t, that's actually what I'm going to be looking for here. So I have, vnaught, which is 10, vfinal, which is 30, and I have the acceleration which equals 2. The last one is Δx. It's the distance over which this thing is accelerated. I actually don't know what that is. All I'm told is the initial and final velocities and the acceleration in between; I don't know the distance. But luckily, I actually have 3 out of 5 variables. I have 1, 2, and 3. So the equation that's going to relate them and also give me time is going to be the first kinematic equation. This is basically just vfinal=vinitial+at. Change of velocity equals acceleration times time. So you rearrange for this because we really want to solve for this time over here, and you'll find that we're going to get vfinal−vinitiala. So we go and plug this in. This is going to be 30 minus 10 divided by 2, and this is going to give me 10, and this is going to be in seconds. So this number here, this 10 seconds, is now what I just plug into the power average equation, and I'm done. So this paverage is just going to be the change in mechanical energy, which is 6,000 joules divided by 10 seconds, and so that's going to give me an average power of 600 watts. Alright. So hopefully that makes sense, guys. Let me know if you have any questions. That's it for this one.
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Power
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