The lazy grasshopper

I was wondering if there is a significance to theta being 45 degrees?

I sometimes dont rely on calculus because sin theta is only between -1 and 1... However even that can be checked by seeing equality condition[1]

Is there a better proof?

I first got lost in the parabola equations and circle equation and the condition for a common tangent etc.... Which could have given the answer but i lost my patience...

First lets see the situation in a diagram..

then we notice that at the point of contact the tangent is the same as teh direction of the velocity ..

that gives us a good start...

Now waht we do is to start our analysis from there

the velocity at that point is u and range is 2 r cos θ

\\2 r\cos \theta = u^2 sin(2\theta)/g \\2 r\cos \theta =2 u^2 sin(\theta) cos(\theta)/g \\r =u^2 sin(\theta)/g \\u^2=rg/sin(\theta)

Now we need to conserve energy to reach the bottom point of the loop
velocity at the bottom is v..
that is found by

\\1/2mu^2+mgr(1+\sin\theta) \\=1/2mrg/\sin\theta+mgr(1+\sin\theta) \\=mgr(1+\sin\theta+1/2\sin\theta)

Now minimum velocity at the bottom is found using some calculus which gives theta to be 45 degrees :),

\\1/2mv^2=mgr(1+1/\sqrt{2}+1/\sqrt{2}) \\1/2mv^2=mgr(1+\sqrt{2}) \\v=\sqrt{2gr(1+\sqrt{2})}

Well sorry Kaymant sir, i could not resist solving this one..
Took me 3 different methods and half an hour to reach the solution.. 2 failed badly.. and I loved this one.. :)