Ideally I would like help with all questions if someone is able and willing to help ( I had to post them all together since they are related and so the expert could better understand). If not, I specifically would like help with the last question (question 33). Thank you very much in advance!

Hi.  I still don't get it.  I'm not sure what sin and cos have to do with it.  But either way, I'm still stuck in the same place.   I don't know how you got from steps 3 and 4 to the next step.  One is for the x-axis and the other for the y-axis.  How did they end up merged together?  If you apply the Pythagorean theorem, you get either:v = sqrt{(vcos) 2 + (vsin) 2}  Then what?  Or you get:v = sqrt{(m1v1/m1+m2)2 + (m2v2/m1+m2)2}   Now what?  That "adding" step just looks like the Pythagorean theorem, with v2 being the c in c2 = a2 + b2, but what is going on on the other side of the equation?   I don't understand how in the denominator you went from:(m1 + m2)2 + (m1 + m2)2 to(m1 + m2)2It seems to me like (m1 + m2)2 + (m1 + m2)2 = 2(m1 + m2)2 Explain as if I didn't understand the algebra, because I think that's the issue here.

Kindly give me a detailed answer in a clear handwriting of all the subparts by mentioning the subpart's name. It is one whole question. All the subparts are related. Take your time solving this. I am attaching the full question. All the questions are related. This is my second time uploading these questions as I got wrong answers previously. So kindly answer this one by following the given instructions. I will upvote if the answer is correct. Thank you.