The 2 formulas we need:
Position as a function of time:
p(t) = 0.5 * a * t^2 + v0 * t + p0
Velocity as a function of time:
v(t) = a * t + v0
We are given the initial speed off the tee, but not the angle, so we don't know the horizontal and vertical components (vx0, vy0).
Use -9.8 m/sec^2 as the acceleration due to gravity.
When the ball is at the apogee the vertical component of the velocity will be 0, so
v(t) = 0 = -9.8 * t + vy0
9.8 * t = vy0
t = vy0 / 9.8
At that same time
P(t) = 24.8 = 0.5 * -9.8 * t^2 + vy0 * t + 0
24.8 = -4.9 * t^2 + vy0 * t
Substituting
24.8 = -4.9 * (vy0/9.8)^2 + vy0 * vy0/9.8
24.8 = -4.9 * (vy0/9.8)^2 + vy0 * vy0/9.8
24.8 = -vy0^2 / 19.6 + vy0^2 / 9.8
486.08 = -vy0^2 + 2 * vy0^2
486.08 = vy0^2
22.047222 = vy0
Because the initial speed of the ball is 52.0,
(vx0, vy0) = (52.0 * cos(p), 52.0 * sin(p))
22.047222 = vy0 = 52.0 * sin(p)
0.423985 = sin(p)
arcsin(0.423985) = p
25.0864 = p
vx0 = 52 * cos(25.0864)
vx0 = 52 * 0.905669
vx0 = 47.095
Now that we know the initial velocity (speed plus direction), the next question is how long does it take to get to the height 6 meters below the apogee?
py(t) = (24.8 - 6) = 0.5 * -9.8 * t^2 + 22.047222 * t + 0
18.8 = -4.9 * t^2 + 22.047222 * t
0 = -4.9 * t^2 + 22.047222 * t - 18.8
Use the general quadratic equation and you get two values for t, 1.14314987 or 3.356283191
We use the first, smaller value, because that represent the time it take to reach the desired height as it travels upward to the apogee.
To find the vertical component of the velocity at that time
vy(1.239563997) = -9.8 * 1.14314987 + 22.047222 = 10.844353274
The horizontal component of the velocity will be vx0 because there is no acceleration in the horizontal direction. Therefore the velocity is (47.095, 10.844). The speed is the magnitude of the vector
s = sqrt(47.095^2 +10.844^2)
s = 48.3 m/s