The Postflight-

K_er= rotational kinetic energy= (1/2)Iw²

U_gym=mgy; TE_flight=(1/2)Iw² +mgy

The hips of the gymnast act as a center of mass and trace a parabolic path characteristic of projectile motion.

The position vectors

X_f=x_i+vx_it=(1/2)a_xt²

Y_f=y_i+vy_it=(1/2)a_yt²

R=total position

R_f=x_f+y_f=r_i+v_it+(1/2)at²

In our case and referring to the figure above

V_xi=v_icosø

V_yi= v_isinø

X_f=x_i+(v_icosø)t+(1/2)a_xt²

Y_f=y_i+(v_isinø)t+ (1/2)a_yt²

A_y=-g

So: y_f=y_it(v_isinø)t+-(1/2)gt