00:02
All right, in this question, we have a block against a compressed spring that's going to be released, collide with another block, and then the pair are going to slide up a hill and we're going to find the final velocity of the two blocks at the top of that hill.
00:16
So we have three basic things that are going on here.
00:19
We have what i'm going to call phase one, where the spring releases block m1, and then we are going to have phase two, you know, actually let me grab a let me grab a lighter blue.
00:39
I think it'll be a little bit easier to see.
00:40
And then we're going to have phase two, which is going to be the collision of m1 with m2.
00:46
And then we are going to have phase three, which is the rising of the pair of blocks up the hill.
00:56
We're going to have to handle all three of these to get this answer.
01:02
So to start with phase one, that's going to be a conservation of energy situation, conservation of mechanical energy.
01:11
So there's elastic potential energy stored in the spring equal to one half kx squared, and that will be equal to one half m1, i'm going to call it v1 squared.
01:26
And i don't like dealing with fractions, so we're going to multiply everything through by two, giving me kx squared equals m v1 squared.
01:38
So to solve for v1, i'm going to divide both sides by m, i'm going to take the square root.
01:48
And since x1 is already squared, i'm going to pull it outside the square root, because the square root of something squared is just the thing.
01:58
And so we have a compression of 0 .05 meters times the square root of 500 newtons per meter, divided by three kilograms.
02:10
And that gives me a velocity of 0 .645 meters per second for block one.
02:19
Now we can handle phase two, this is going to be a conservation of momentum, because it's a collision, right? so the initial momentum before the collision equals the final momentum after the collision, our initial momentum is carried entirely in m1, so m1 v1 equals, and they are sticking together, so we have to add their masses m1 plus m2, multiplied by, i'm just going to go ahead and call it v.
02:51
So the velocity of the blocks together after the collision will be equal to m1 v1 over the sum of the masses.
03:00
So we will take that three kilograms times 0 .645 divided by three plus four kilograms, and i get 0 .276 meters per second.
03:19
And now finally we're going to go back to an energy approach.
03:24
How do i know that? because we have something that's going from moving to elevated and possibly moving.
03:32
So we're going to have kinetic energy of the blocks at the bottom of the ramp equals the gravitational energy plus the kinetic energy at the top...