Here, the initial height of the stone off the ground is s0=0; the acceleration is g (the acceleration due to gravity)=-32.2 ft/(sec^2); and final height of the stone is 81 ft. Substituting these values into the above equation gives us
81 ft = 0 ft +(1/2)(-32 ft/(sec^2))(t^2) + v0(t). We must solve this for the initial velocity, v0. Unfortunately, we don't yet know how long it will take for the stone to reach its max height, so both t and v0 are variables here.
81 ft = -16 ft/(sec^2)(t^2) + v0(ft/sec)(t)
Gravity will slow the upward progress of the stone: v = v0 - 16 (ft/sec)(t). Solving this for t:
0 = v0 - 16 (ft/sec)(t) => t = v0/[-16(ft/sec^2). Substitute this equation for t into the previous expression and then solve the resulting equation for the initial velocity, v0.
Alternatively, use the following formula, which does not involve time, t:
v^2 = v0^2 + 2 a(s).
Using v=0, v0 unknown, a = g = -32.2 ft/(sec^2), and s = 81 feet, solve this for v0.
