Question

With what initial velocity must an object be thrown upward from a height of 2 meters to reach a maximum hieght of 200 meters

Answer 1

The equations of motion to use for a constant acceleration a = g = -9.81:
$v = gt + v_0$
$y = \frac{1}{2} g t^{2} + v_0t + y_0$

at the highest point the velocity must be zero:
$0 = gt + v_0 \\ v_0 = -gt$
combine both equations:
$y = \frac{1}{2} \frac{v_0^{2} }{g} - \frac{v_0^{2} }{g} + y_0= -\frac{1}{2} \frac{v_0^{2} }{g} + y_0$
Solve for v₀:
$v_0 = \sqrt{2g(y_0 - y)}$