Vectors and Projectiles Legacy Problem #32 Guided Solution

A good problem solver takes the time to read the problem carefully to get a good mental picture of what's going on, to write down the things that they know and the thing that they're looking for, and to plot out a strategy for how to get from the known information to the unknown information. In this problem, we've got a picture of a girl on a swing, I love you daddy, who jumps out of the swing at an angle of 30 degrees to the horizontal at a known speed of 2.2. She goes through the air and she lands 1.09 meters horizontally from the launch location. We don't know whether this is at the same height or not, and likely it isn't if you're getting the picture of a girl jumping out of the swing. Usually she leaves the swing from an elevated position and she usually lands on the ground, so we would reason that this is not a symmetrical trajectory that she lands at a different height than she started. Now we're going to write some things down here, we're going to organize it, I suggest you do it in a so-called XY table, where you're listing horizontal information in one column and vertical information in another column, and one of the first things you want to list is DX equal 1.09 meters. Then you get to this 2.2 meters per second, and that's VO. It's not a VOX, it's not a VOY, it's VO and it's information you can use to calculate VOX and VOY along with the 30 degree angle. The so-called Voxboy equations that you might find on the overview page here can be used to calculate VOX, VOY, VOX is just 2.2 times the cosine of 30 degrees, and I get 1.9053 and you can round it if you want, but make sure that you know all the digits, you're going to use all those digits throughout the remainder of the solution of the problem. And then when I do the VOY, I'm going to go 2.2 times the sine of 30 degrees, and that's 1.100 exactly. And that's the answer to my question A. Now once I get this VOX, my VOY listed in my table, and now I should think about what else I know, and one of the things I know is AX, and I know AY, I know that for all projectiles, AX is zero meters per second and AY is negative 9.8 meters per second. So if I list that in my table as well, you might immediately recognize that in the X column, in the horizontal information column, you have three pieces of information, and with three pieces of information, you calculate a fourth and a fifth, or however many you need to calculate. And the one thing that will be of interest here, essentially the part B question, is the time that olive is in the air. And that can be found if you use the only equation we typically ever use for X information, that's the one that goes DX equal VOX times T. You can take the 1.09 meters and say that's equal to 1.9053 times T, and you can solve for T, and when you do, you'll get a T in this problem of 0.5721 seconds. So that's the time to go up, come back down, and finally fall a little further and land on the ground, 0.5721 seconds. And what they're asking you to calculate is what's the vertical height from which olive jumps from the swing? Well, they're asking you really to calculate DY, relative to where she lands on the ground, how high up is she, is the same question as asking how far did she fall from her starting location to the ground? Calculate DY. So we have an equation on the overview page for DY that goes something like this, DY equal the original Y times T plus 1.5 AT squared. And if you look at that right side of the equation, we know everything we need to know to plug in there. We know the original Y is 1.1000. We know T is 0.5721 seconds, and we know AY is negative 9.8 meters per second squared. And so now plug all those numbers in, and be careful of your minus sign there on the negative 9.8. You're going to get a positive term, the first term will be 0.6293, and the second term will be negative 1.6038, and it comes out to be a DY value of negative 1.03 meters, which tells us that if she fell 1.03 meters, then the height from which she jumped out of the swing had to have been 1.03 meters.