 We have a graph of velocity one and two. That was our cart which had stuck together with the second cart versus velocity one. We've made our graph, we found our line of best fit, we found our slope, we did all that on our calculator last class. Now I'm going to show you how you can take the slope and from that work out with the mass of that second car was. And to do that it's going to seem kind of strange what I'm going to do is I'm going to go back in time with you a little bit. And I want to think back to some of the stuff we used to do with graphs back in in physics 20. So back in physics 20 or even science 10 you would get a displacement versus time graph. Maybe it was just a really simple one it looked like this. Nice linear graph. And if you found the slope of that graph what did it give you do you remember? Good it gave you velocity. Now I don't know if your physics 20 teacher or science 10 teacher told you why that was. Maybe it's just magic or maybe there's a reason for it. There's a reason for it. Okay here's the reason why that slope is velocity when you're talking about a displacement time graph. Slope in general is rise over run. Now if you apply rise over run to this specific graph the rise of the graph is displacement and the run of the graph is time. And so what you end up getting here in the slope formula is like a little piece of algebra I usually call it a little piece of algebra which just happens to match up with a physics variable we know about. So do you know of an equation that has d over t on one side? Which which is it? It's v equals d over t right off your formula sheet. So because we have this little piece of algebra which is really our slope it's our rise over run but it's equal to another physics constant we're able to sort of work out that the slope of this graph gives velocity and that one was easy enough to do because the formula that you got from doing rise over run looks exactly like it did in your data sheet. You didn't have to think too hard about it or manipulate it or anything. Well we're going to do the exact same thing here today. We're going to work out what the slope of our graph is and we're going to see what little piece of algebra that's equivalent to. The only thing that's going to be different is when I asked you what's d over t and all of you sort of knew oh that's velocity it'll be a little harder to figure out what our slope is equivalent to but we can still do it. Okay so here we go first things first I'm going to write down what is our slope of this graph. So the slope m is rise over run. Our rise is v1 plus 2 that's our rise variable and our run is the v1. So there we go that's what our slope is equivalent to but the problem is is there's no formula on the formula sheet that says v1 plus 2 over v1 that we can just match it up really easily but that's okay we can make a formula. We're going to see if we can put together a formula knowing what we know about the conservation of momentum and see if in that formula we can manipulate it until we get v1 plus 2 over v1. So let's do a little conservation of momentum formula like we did last class. We start off kind of always in the same way. We say the sum of the initial momentum equals the sum of the final momentum. All of the objects that were moving and had momentum to begin with if we add together their momentum that total has to be the same as the total of all the objects after they hit what their momentum is. Now in this question before collision only one object was moving that was that cart one and so I could write for that objects momentum mass one velocity one and after the two objects hit they had some velcro on them so they stuck together. Do you remember what we used to do used to do last class when we would have objects stick together? What would you do with their masses? You'd add them together. So I'm going to write mass one plus mass two and then the velocity we're just going to call velocity one plus two just like before because the two objects stuck together. It was a hit and stick collision and they moved off with one velocity. This looks just like your notes. This is the same formula we used for a hit and stick sort of collision. Now we can do some substituting because we do know what mass one was. It was 209 grams. No need to convert into kilograms. It's totally fine to leave that as grams but I'm not going to substitute in v1. I'm going to leave that just as the variable. Mass one 209 grams substitute that in again. Mass two is the answer to the lab. That's what I want to solve for. I want to know what that second mass was and v1 plus two I'm also going to leave as an unknown. That probably seems like a really bad idea because we have three unknowns so we can't really solve this formula as is right now. But what I can do is I can rearrange the conservation momentum formula and watch how I'm going to rearrange it here. I'm going to move the 209 grams plus m2 over here. So 209 grams plus m2. I move it to the left hand side of the denominator. So I divided both sides of the equation by 209 grams plus m2. That just leaves me with v1 plus two on the right. And then I'm going to divide both sides by v1 and I'm going to move that to the other side like that. All right so it probably seems totally weird and random to rearrange the formula like that because you're thinking this doesn't help me solve for m2. But we can kind of use that same little trick we did back in physics 20 science 10 again. We now have that little piece of algebra I like to call it. The little piece of algebra which is equivalent to something back in science 10 physics 20 d over t which was the slope was equivalent to velocity. Now I have the slope again. Here's that same slope. I found that same slope again in my algebra. And since I already know the slope is the same as the number 0.41 what I can do is substitute in and solve. So 209 grams over 209 grams plus m2 equals 0.41. And the reason I could substitute in v1 plus two over v1 with 0.41 is because 0.41 is the slope of the line. And we know the slope of the line is the same as v1 plus two over v1. So now I can just go through and do a little bit of algebra. I'm going to have to go down to where my evaluation was here. Sorry about that. So I'll get 209 grams equals 0.41 times 209 grams plus m2. I'll divide both sides by 0.41. So let's see what that gives me. 209 divided by 0.41 is 509.7561 is equal to 209 grams plus m2. And then I'll just subtract 209 from both sides. So I get that mass 2 to 2 sigdigs works out to 3.0 times 10 to the 2 grams.