We have three balls which are thrown up at the same velocity V not, but they're thrown a different angles, one of them straight up. One of them's at a little bit of an angle from Vertical Williams at a large angle from vertical. And we want to know if they're thrown at the same speed, but at different angles. When they reached the same height indicated by this green line, which of thumb will have the smallest speed. So to answer this, we need Thio use conservation of energy So initially, there at a height of zero so they only have kinetic energy. 1/2 V I squared or V not squared. Let me write that down. Vino is equal to V I Just because I started writing it like that on Ben when they reach this height, they're gonna have some gravitational potential energy as well as some kinetic energy. So first thing we can do right off the bat to find the final velocity is actually divide both sides by M. Since all of the terms here have amassed in them. So then we get 1/2 V. I squared is equal to um is equal to sorry. No mass year because they'll cancel it. G h final, which is the height that they all all three of them reach at this point in time. Um, plus one half V f squared. So we want thio rearrange for VF. So, um, that gets us 1/2 The squared is equal to 1/2 V. I squared minus G h f. Now, this is actually enough to tell us which of them will be the slowest. But let's just to make things super obvious, multiply both sides by 1/2 Um, or multiply both sides by two. Rather so V squared is equal to V. I squared minus, um, to G h f and then take the square root of that two finds the final velocities equals square root of V. I squared minus two. She h f So what does the final velocity depend on? Um, G is a constant here. So all that the final velocity of a ball depends on is the initial velocity and the height that it reaches. So, Theo initial velocities were given are all the same and the heights that they reach, Um, you just label that, um h f is this dotted line. The heights they reach are also the same. So all the variables on the right hand side of the equation of the same for all three of the balls eso it does not depend on the angle of the velocity as long as the initial velocities are the same. There's also no mass term here because we canceled out maths in the very first step. So does not depend on Mass. So that tells us that the answer is, See, all three balls have the same final velocity because it doesn't depend on the angle of the initial velocity as long as those are the same and it doesn't depend on mass because we can't sell those out.