Why is $$h = v sin(\alpha) b$$, I know that h is a constant $$ = \frac{d\theta}{dt} r^2$$ but in every problem h is taken as the product of the starting velocity and the position and the sine of the angle of projection.
In the same problem , $$\frac{du}{d\theta} = - \frac{1}{h} \frac{dr}{dt}$$ = $$- \frac{cot(\alpha)}{b}$$
, I couldn't quite get why the cot part is true
In such classical mechanics problems, h happens to be the angular momentum per unit mass (usually taken about the origin) and it remains conserved for Kepler problems. So to compute the angular momentum per unit mass at the point ($b,0$) about the origin, you multiply the component of the velocity perpendicular to the x-direction ($v_0 \sin \alpha$) with the distance from the origin ($b$).