First, we need to find the time it takes for the ball to hit the ground. Since the ball is thrown horizontally, its vertical motion is only due to gravity, and we can use the kinematic equation:
y = yo + voy t - 1/2 g t^2
where y is the vertical position, yo is the initial vertical position (50.5 m), voy is the initial vertical velocity (zero since the ball is thrown horizontally), g is the acceleration due to gravity (-9.8 m/s^2), and t is the time.
We want to find the time it takes for the ball to hit the ground, so we set y = 0 and solve for t:
0 = 50.5 + 0 t - 1/2 (9.8) t^2
t = sqrt((2*50.5)/9.8) ≈ 3.2 s
Now we can find the horizontal distance the ball travels in 3.2 s, using the equation:
x = vox t
where x is the horizontal distance, vox is the initial horizontal velocity (17.9 m/s), and t is the time we just found:
x = 17.9 m/s * 3.2 s ≈ 57.3 m
Therefore, when the balloon lands on the ground, the girl is about 57.3 meters away from where the ball landed.