A rocket is launched straight up with constant acceleration. Four seconds after liftoff, a bolt falls off the side of the rocket. The bolt hits the ground 6.0 s later. What was the rocket's acceleration?

Video transcript

Everyone in this problem. A hot air balloon released from rest in a meadow moves vertically upward with a steady acceleration. After nine seconds of motion, a stone stuck on the bottom of the basket falls down and strikes the meadow. Seven seconds later, we're asked to calculate the acceleration of the hot air balloon. All right. So let's think about this. Okay. We have a steady acceleration. So we know that we can use our you am equations. Okay. Uniformly accelerated motion. We have a steady acceleration so we can use those equations which are also our kid a Matic equations. If your professor calls them by that name and we have two things to consider. We have the hot air balloon and we have this stone that falls from the basket. So let's start with the hot airport, Its initial speed once its initial speed while we're told it's released from rest. So its initial speed or velocity is 0m/s. The final speed, we don't know the acceleration is what we're trying to figure out. Okay. The acceleration of the hot air balloon the time. Well, let's think about this nine seconds. Okay. Let's consider nine seconds of motion and then the stone falls. So let's consider the time to be nine seconds and we don't know the height of the hot air balloon. Alright, so we have two values that we know, which is not enough to use our equations to find the acceleration. So let's go over to the stone and see if we can fill in some of these gaps. Now, the initial speed or velocity of the stone that's going to be the final speed and final velocity of the basket, the stone is stuck to the basket. So as soon as that stone falls off and start its separate motion, it's going to be traveling at the exact same speed as the basket. So this is going to be equal to V F. Now, the final speed of the stone, we don't know The acceleration and the stone is just falling. So the acceleration is gonna be the acceleration due to gravity. If we take upwards to be our positive y direction, then the acceleration due to gravity is going to be negative 9. meters per second squared. The time that the stone is falling is seven seconds. And what about Delta Y? Well, the distance that the stone is going to fall, it's going to be the exact same as a distance. E hot air balloon rose. Okay. The hot air balloon started at the ground rose to some height. Delta Y, the stone is gonna fall off at that height and then fall back to the ground. Okay. So delta Y s the magnitude of these values is going to be the same as Delta Y. It's just going to be negative, okay, because it's traveling in the opposite direction, the balloon was going upwards, the stone is falling downwards. Alright. So now we have a relationship between the speed of the stone, the speed of the hot air balloon and we have a relationship between the distance that they travel as well. So we don't have enough values to figure out a in the hot air balloon, but we do have two values that are related. We can write an equation for the hot air balloon and the stone that's gonna be two equations with two unknowns and we'll be able to solve. So let's start with the hot air balloon. Now, for the hot air balloon, we want to take the equation without the acceleration. A first. We don't know a and we don't have a relationship between A and the stone. So we want to consider the equation without a first. So that's gonna be delta Y Is equal to 1/ V naught plus V F times T. If we fill in what we know we have delta, why Is equal to 1/ times while the initial speed is zero. So it's just gonna be times V F times the time, which is nine seconds. Alright. So that's as much as we can do here. Let's work with the stone. Now, for the stone, we also want to have an equation with delta Y and V F. Okay. So what we're gonna do for the stone is we're going to choose an equation that doesn't include the final speed. Let me go up here. Okay. The final speed of the stone, we're gonna include the initial speed of the stone, which is the final speed of the hot air balloon. And if we do that, the equation is going to be delta Y of the stone is equal to V naught of the stone times the time T plus one half A T squared. And this is the acceleration of the stone. And all of these times are because of the stone. Okay. We're looking at the stone. Now delta Y for the stone, we know that this is gonna be equal to negative delta Y. We know that V not S is going to be equal to V F. We have V F times a time of seven seconds plus one half the acceleration of the stone negative 9. m per second squared Times the time squared, which is gonna be seven seconds squared. So this tells us a negative delta Y it's going to be equal to via times seven seconds, one half times negative 9.8 times seven squared. This is going to be negative 240.1, the unit of second squared world cancel. And we're left with just a unit of meters. So now we have an equation where delta Y is equal to some value. And we have another equation where negative delta Y is equal to some value. So we can substitute one into the other. So let's call the equation that we found for the stone. Negative delta Y is equal to V F times seven seconds minus 240.1 m. We're gonna call that equation star and we're gonna substitute that into our equation. Delta Y is equal to one half V F times nine seconds. Now we get that negative VF times seven seconds is equal to whoops plus one plus 240.1 m Is equal to 4.5 seconds times. VF Okay. We had nine seconds times a half that gives us 4.5 seconds. Now we only have VF as are unknown we can solve for V F. Okay. We're gonna move negative VF time seven seconds to the right hand side. We get 240.1 m is equal to 11.5 seconds times. VF we divide by 11.5 seconds to isolate VF and we get that V F is equal to 20.878 to 6 m per second. Kr units were meters divided by seconds. We have meters per second and this is the final speed of our hot air balloon at the point where the stone falls from it. Now, we have to remember that's not what we're looking for. We go back up, we're looking for the acceleration A. We have Vienna, we have T and now we have V F. So now we have three known quantities. We can use our um equation to find the acceleration that we were looking for. So we're gonna choose the um equation without delta Y. We don't have information about delta Y And that's gonna be the following VF is equal to be not plus 80. The V F we found was 20.878 to 6 m per second. V not is zero m per second. Our acceleration a times the time nine seconds. Okay. If we divide by nine seconds, we get an acceleration of 2. m per second squared. And this is a positive value which makes sense. We know our hot air balloon is going upwards. So the acceleration should be positive to indicate that upwards motion. Alright. So if we look back at our answer traces, we found an acceleration About 2.3 m/s squared. That corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.