Power of Pushing a Box

by Patrick Ford
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Hey everyone, Let's check this out here. So, we have a 15 kg 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 m per second, but it's accelerating over some distance. And then at some point over here it's final velocity is going to be 30. We're told that the acceleration along this interval here, this A is equal to two. That's basically all we know in the first problem, we want to calculate what is the change in the mechanical energy of the blocks. Mechanical energy. So, the variable that we're looking for is delta M. E. It's the change in mechanical energy. Now, remember for blocks, mechanical energy 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. All right, So, let's consider each one of these terms here. Now, we have a block that's not attached to any springs. There's 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 bloc's kinetic energy is changing across this interval and because it's going faster right from 10 to 30 and therefore there's a change in the mechanical energy. So that's delta Emmy. This is just gonna equal K final minus K. Initial. So delta Emmy here is just gonna equal the formula for kinetic energy is one half M. V final squared and then one half mv initial squared. Now I have all of those values, I have the masses and velocities. So I can actually figure this out. My delta Emmy. It's just gonna be one half of 15 times V final which is squared minus one half times 15 which is times squared when you go ahead and plug this all into your calculator, you're gonna get 6000 jewels. So one way you can also think about this here is that there's a force that is pushing this thing that's pushing this block, that's why it's causing it to accelerate. We don't know what that force is but that forces 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 6000 jewels. Alright, 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 two 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 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 gonna 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 velocity, the acceleration and the final velocity. These are Kinnah Matic variables. So if I want to figure out t which is also a kingdom attics variable. I just need to do exactly what I did when we talked about one dimensional motion. I need to set up my five variables and I need three out of five. So I don't I don't know t that's actually what I'm gonna be looking for here. So I have venus which is 10 v final which is 30 and I have the acceleration which equals to the last one is delta X. It's the distance over which these things is accelerated. I actually don't know what that is. All I'm told is the initial and final velocities and the acceleration between I don't know what the distance, but luckily I actually have three out of five variables, I have 12 and three to the equation that's going to relate them. And also give me time. It's gonna be the first cinematic equation. This is basically just the final equals v knots plus 80. Change of velocity equals acceleration times time. So you re arrange for this because we really want to solve for this time over here and you'll find that we're gonna get the final minus the initial divided by a equals time. So we gotta plug this in. This is going to be 30 -10 divided by two. And this is going to give me 10 and this is gonna 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 p average is just gonna be the change in the mechanical energy which is joules divided by 10 seconds. And so that's gonna 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