A slingshot will shoot a 10-g pebble 22.0 m straight up. (b) With the same potential energy stored in the rubber band, how high can the slingshot shoot a 25-g pebble? (c) What physical effects did you ignore in solving this problem?

Video transcript

Hey, everyone in this problem, we have a 2.5 grand fall fired upward using a spring loaded toy gun. The ball goes only five m above its initial position. How high can a 10 grand ball be launched by a toy gun having the same potential energy as the spring? Alright, so we're asked to find The height that a 10 grand ball can reach if it has the same potential energy as the spring. So let's consider our conservation of mechanical energy. We know that the kinetic energy initially plus the potential energy initially, it's going to be equal to the kinetic energy final plus the potential energy. Now we're going to consider this for that first spring loaded toy gun. Now, when we talk about our potential energy, we know we have the potential energy from the spring. That's what the question is asking us about. But we also have to consider the potential energy because of gravity. We're shooting this ball straight upwards. And so there's also going to be potential due to gravity. And so we have our kinetic energy plus our potential energy due to gravity initially plus our potential energy due to the spring initially equal to the kinetic energy at the end plus the potential energy due to gravity at the end plus the potential energy through the spring at the end. Alright, so we're considering this initial case is pre launch and the final cases at max height. Okay. So this is going to be corresponding to maximum height of the ball and the left hand side corresponds to pre launch, no pre launch and at maximum height, the speed is going to be zero. Okay. Before we launch this ball, it's just sitting inside of the gun. So it's gonna have no speed. And at the maximum height, the ball is going up, up, up, up, up, it reaches its maximum height and it momentarily stops before it turns around and comes back down. Okay. So the speed is zero there too, which means that the kinetic energy K naw and KF both are going to be zero. All right. Now, if we let the initial position be our reference point. Okay. So the initial position has height H And the gravitational potential energy before the launch is also zero. So on the left hand side of our equation, we only have our spring potential energy on the right hand side of our equation. Okay. We only have our gravitational potential energy because once that spring loaded toy gun has fired the ball, the spring potential energy goes to zero. Okay. So we're converting all of that spring energy into height into gravitational potential energy. And so our spring potential energy is going to be equal to the mass of the first fall gravitational acceleration times the final height of that first ball. This is 2.5g, the mass we're going to convert that to kg. So we're gonna multiply by one kg Per 1000g, Gravitational acceleration 9.8 m/s squared in the final height while we're told it reaches five m. Alright. So if we work this out on our calculator and we have kilogram, the unit of Graham cancels out meter per second squared times meter. This is going to give us a unit of a jewel And we get 0.12, jewels as our spring potential energy. So that's great. We found that spring potential energy. But remember what we're trying to find is the height of 10 grand ball could be launched using that same potential energy. Alright, so this was our initial case with the first ball. What if we have the second ball? Okay. So we know that the spring potential energy and in this case of ball to initially is going to be converted completely in the gravitational potential energy after the ball is fired. All right. Well, we know that our spring potential energy is 0.1225 jewels. Now it's the same spring potential energy as with the first fall, our gravitational potential energy is M2. Okay. The mass Of the second ball. I'm gonna write that here. M two G times the height that it reaches the second ball. So we have 10 g. And again, we're gonna convert two kg, we multiply by one kg per 1000 g in the unit of Graham cancels Gravitational acceleration 9.8 m/s squared and age to that value we're looking for. All right. So we wanna isolate for H two. When we do that, we're gonna have that H2 Is equal to 0.1, 2, 2, 5 jewels divided by 10 g, times one kg per 1000 g Times 9.8 m/s squared. Okay. All that's in the denominator. And if we work this out on our calculator, we get a height of 1.25 m. And so the height a 10 g ball can reach using that same spring potential energy from our toy gun is going to be 1.25 m. If we go back up to our answer choice, we see that that corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.