I can do a few
a) The velocity is positive on 0 < t < 1
This means the ball is rising during this time
b) The antiderivative is s (t) = 32t - 16t^2 + C
At s(0) we have 32(0) - 16(0)^2 + C = 0
So C = 0
c) s(1) - s(1/2) = [ 32(1) - 16(1)^2 ] - [ 32(1/2) - 16(1/2)^2 ] = [ 16] - [ 16 - 4] = 4
This represents the change in postion from t = 1/2 to t = 1
[ It rises 4 ft in this interval = the displacement on this interval ]
d) At t = (1/2).....the height of the triangle formed is 32 - 32(1/2) = 32 - 16 = 16
So....the area = (1/2) base * height = (1/2) (1/2)(16) = 4
This area also represents the displacement on t =1/2 to t =1
f) s(2) - s(0) = [ 32(2) - 16(2)^2] - 0 = 0
This means that the total displacement = 0 ......it falls as far as it rises
Note that the velocity at t = 0 = 32 ft/s
And at t = 2 = -32 ft/s
This makes sense.....the terminal velocity = the beginning velocity....only in the opposite direction