A rocket starts from rest and moves upward from the surface of the earth. For the first 10.0 s of its motion, the vertical acceleration of the rocket is given by ay = (2.80 m/s3)t, where the +y-direction is upward. (b) What is the speed of the rocket when it is 325 m above the surface of the earth?

Video transcript

Welcome back everybody. We are taking a look at a hot air balloon and we are told a couple of different things. We are told that the hot air balloon is initially at rest and once it is released starts moving upwards now during the 1st 70 seconds of time, it's vertical acceleration is defined by this function of time .008 T. Now we are tasked with finding what the vertical velocity is when the height is six m off the ground. So here's how we are going to do this. We're gonna need a couple equations here. Equation one that we want to find is first our equation for our vertical velocity as a function of time. Since we're giving our acceleration in our initial velocity, we can do this, we can say that our initial velocity plus the integral from zero two T. Of R A Y D. T will be our equation for vertical velocity. But we're still going to have to plug in a value to that and the value we're gonna have to plug into that is time. But how are we going to figure out time? Well, we know that at a height of six m that there's going to be a certain velocity. So we can use that to our advantage to solve for time. We know that height is the integral from zero T of our vertical velocity as a function of time. And since we know our height will be able to solve for time using that second equation. So first and foremost, let's start with part one here. We need to find our equation for V. Y. Of T. Is just equal to our vertical velocity which is zero. So nothing will be there. Plus the integral from zero to T. Of this formula right here. 0.8 T. D. T. Which when you plug into a graphing calculator, right? You can get 0.4 T squared. All right now let's go ahead and find our second equation here. We have that our height is equal to the integral of zero T. Of the equation that we just found. 0.4 T square D. T. Which is equal to 1/ he cubed. And I'm just going to move this over a little bit because we are going to want to manipulate this function. Just the tiniest bit. Reason being is because we know what our height is. So we can set this entire thing to our height of six m. And then if we just solve for T we will have the time that we can plug into our equation one right here. So let's do that. I'm gonna multiply both sides by 750. You'll see that these terms cancel out right here and then I will take the cube root of both sides of my equation. These powers cancel out and we are left with T. Is equal to the cube root of six times 750. Which when we plug this into our calculator, we get that T. Is equal to 16.51 seconds. Now we are ready to find our vertical velocity at that height. We are just going to plug in this value into our equation one here, so we have that are vertical velocity at a time of 16. seconds is equal to 0.4 times 16.51 squared. Giving us a final answer of one point oh nine m per second, which corresponds to our final answer choice of C. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.