At an angle $\theta$ with horizontal, a particle is projected from surface of the earth of radius R with considerable large velocity $V_0$ to attain a maximum height ‘h’. Then

speed of the particle at the maximum height $\frac{R{V}_o\cos \theta }{(R+h)}$

speed of the particle at the maximum height is $\frac{R{V}_0\tan \theta }{\sqrt{2}(R+h)}$

for the particle projected from surface of the earth, angular momentum is conserved about centre of  earth

for the particle projected from surface of the earth, angular momentum cannot be conserved about  centre of earth due to the gravitational force of attraction acting on the body due to the earth.