A 2.00-kg box is moving to the right with speed 9.00 m/s on a horizontal, frictionless surface. At t = 0 a horizontal force is applied to the box. The force is directed to the left and has magnitude F(t) = (6.00 N/s2)t2. (a) What distance does the box move from its position at t = 0 before its speed is reduced to zero?

Video transcript

Welcome back everybody. We have a block that is sliding on a frictionless surface. So let me go ahead and draw that out real quick. And we are told a couple of different things. We are told that the mass of this block is eight kg And at time equals zero. A couple of things are happening here, right? So at time equals zero. This box is currently sliding to the left With an initial speed of 12.8 m/s. I'm gonna designate the right direction as the positive x axis. So meaning the left direction is negative. So I'm gonna make this velocity negative here. We are also told that at this very instance a force is introduced. Now, this force is given as a function of time. We are told that the force is 5.00 newtons per second squared times our times square. And we are asked to find the displacement of this block when our final velocity is equal to zero. May seem like a lot to unpack. But let's just take it one step at a time here. Well, in order to find our delta X, we are first going to need our position as a function of time, right? Because we're dealing with a conditional that deals with time. But what is this going to be. But we know a couple of things here. We know from Newton's second law that our force is equal to our mass times acceleration. And the reason I bring that up is we are looking at finding a function of time. And the only other function of time that we're given is this force. So we're gonna have to use that in some way, shape or form. We are also told or no that the integral of acceleration with respect to time is equal to velocity as a function of time. And the integral of velocity with respect to time is equal to our position with respect to time. So this is what we're looking for right here and we're gonna have to use all these laws to find it. But let's just like I said, take it one step at a time. So first and foremost to get to here, we are going to need to find our acceleration which we can use Newton's second law to find. So let's go ahead and do that. We have that our force is equal to our mass times acceleration. And since force is given as a function of time, we're gonna have our acceleration as a function of time. I'm going to divide both sides by our mass and that is going to yield that our acceleration as a function of time is equal to our force divided by our mass. And were given both of those things are force is 5.0 t squared Divided by eight kg. Giving us our acceleration as a function of time which is 0.625 E squared great. Now that we have that let's go ahead and perform. These integral is right here to find our position with respect to time. The first and foremost. First step is going to be the integral of our acceleration with respect to time. This of course is going to be equal to our velocity as a function of time. So this is the integral of this function right here, 0.625 t squared. This gives us 0.625 divided by three times T cubed Plus some constant. Since we do not have bounds on this integral right here. But what is this concept? Well, this constant in this case, since we're dealing with velocity is going to be the constant velocity that we are given, which is our initial velocity. This is equivalent to 0.625 divided by three times a cubed -12.8. Great. So we have our velocity. Let's go ahead and take the integral of that with respect the time to get our position. This is going to be the integral of this function right here. 0.625 over three times. He cute -12.8. Now this is equal 0.6-5 over 12 times. T to the 4th -12. plus some constant C. Because once again are integral does not have bounds. Well what is R. C. This time when we weren't necessarily given a initial position constant, I'm just gonna say that our block started at the origin, which makes sense because I said that going to the right is positive and going to the left is negative. So it can only start right there in the middle at the origin. So our C. Is equal to zero. That makes things pretty easy. So now we have our position as a function of time. But what t do we plug in here? Well if you remember we need to find our position when our final velocity is equal to zero. Well we have velocity as a function of time. So let's set that equal to zero and then solve for our tea. We are looking for T here. So This is going to be 0.625 over three times. T cubed minus 12.8 is equal to zero. I'm gonna add 12.8 on both sides. And this is going to yield the equation 0. to 5/3 times. T cubed is equal 12.8. multiplying by the reciprocal of this fraction on both sides, we get that e cubed is equal 12.8 times three Over 0.625. Finally to get T by itself, we're gonna take the cube root Both sides giving us that our T. is equal to 3.95 seconds. Great. So now we have the time when our final velocity is equal zero. That was the condition that we are given. So now let's go ahead and find the position evaluated at that time. And we are going to be using our position function that we found. So x evaluated a 3.95 seconds is equal to 0.625 divided by 12 Times 3.95 to the power of 4 -12.8 times our time of 3.95. When you plug this into calculator, you get that. This is negative 37.8 m. Alright, so now let's go ahead and find our total change in distance or our displacement. This is going to be equal to the absolute value of our final position minus our initial position. And we have both of these here we have that our final position is negative 37.8 m. We have that our initial position. Well we said that that was zero minus zero. So it's the absolute value. This this is just equal to the absolute value of negative 37.8 m. Which is equal to 37.8 m corresponding to our answer choice. D Thank you guys so much for watching. Hope this video helped. We will see you well in the next one