Hi. I've been trying to solve the first part but I guess I am just terrible at calculus :(
x1 = r*cos(ang)
x2 = r*sin(ang)
v1 = d(x1)/dt
v2 = d(x2)/dt
as you said.
This means v1= -(dr/dt)*sin(ang)*d(ang)/dt?? And likewise v2=(dr/dt)*cos(ang)*d(ang)/dt?
and V=v1+v2=-(dr/dt)*sin(ang)*d(ang)/dt+(dr/dt)*cos(ang)*d(ang)/dt since its the vector sum of radial and circumferential components, right?
But when I do my calculations, I get v1^2+v2^2=(dr/dt)^2 * (dAng/dt)^2 and when I calculate v^2 I get (dr/dt)^2*(dAng/dt)^2*(1-2sin(ang)*cos(ang). They are not the same :( Nor do they give me v^2=(dr/dt)^2 + (r*dAng/dt)^2 like the book says.
I feel like this should be easy but its not. Sorry!