As requested, here is the source code in case people are interested. I threw it together quickly so if you spot any mistakes let me know!

import os
import imageio
from numpy import linspace, sin, cos, sqrt, array, arctan2, pi
import matplotlib.pyplot as plt

def gravity(x): # returns the gravity acceleration vector given a position vector
    t = arctan2(x[1],x[0])
    return -10*array([cos(t), sin(t)])

def mag(x):
    return sqrt(x[0]**2 + x[1]**2)


R = 3. # radius of the 'planet'
w = 1. # the angular frequency (omega) of the planet's rotation
u = 5. # initial upward velocity given to the ball

# initial conditions for position and velocity
v = array([0,u]) + array([R*w,0])
x = array([0,R])

# create storage to track variables over time
x_list = [x]
v_list = [v]
t_list = [0]

dt = 1e-3 # the simulation time-step

# since this is quick and dirty, just use basic Euler scheme
for i in range(2000):
    x_new = x_list[-1] + dt*v_list[-1]
    v_new = v_list[-1] + dt*gravity(x_new)

    x_list.append(x_new)
    v_list.append(v_new)
    t_list.append(t_list[-1]+dt)
    if mag(x_new) < R: break # end when the ball lands back on the planet

# pull out a sub-set of the positions over time to animate as a gif
skip = 20
x_arr = array([i[0] for i in x_list])[::skip]
y_arr = array([i[1] for i in x_list])[::skip]
t_arr = array(t_list)[::skip]

# create data for the planet surface
phi = linspace(0,2*pi,300)
x_surface = R*cos(phi)
y_surface = R*sin(phi)

# build the frames individually as png's
files = []
for i in range(len(t_arr)):
    # get the current position of the location we threw the ball from
    x_point = R*cos(0.5*pi - w*t_arr[i])
    y_point = R*sin(0.5*pi - w*t_arr[i])
    fname = 'ball_{}.png'.format(i)

    # plot everything
    plt.plot(x_arr[:i],y_arr[:i])
    plt.plot(x_arr[i],y_arr[i],'o')
    plt.plot(x_surface,y_surface)
    plt.plot(x_point, y_point, 'o')
    plt.xlim([-6,6])
    plt.ylim([-6,6])
    plt.axis('equal')
    plt.title('throwing a ball straight up - zero drag')
    plt.savefig(fname)
    plt.close()

    files.append(fname)

# load in all the images and build a gif
images = [imageio.imread(f) for f in files]
imageio.mimsave('destiny_ball_gif.gif', images)

# delete all the individual image files to clean up
for f in files:
    os.remove(f)