Calculus Question

d²s/dt² = -9.8
d(ds/dt)/dt = -9.8
d(ds/dt) = -9.8dt
∫d(ds/dt) = ∫-9.8dt
ds/dt = -9.8t + C1 ...... [0]

ds/dt = -9.8t + C1
ds = (-9.8t + C1)dt
∫ds = ∫(-9.8t + C1)dt
s = -4.9t² + C1t + C2 ...... [1]

Put t = 0and s = 0 into [1] :
0 = 0 + 0 + C2
C2 = 0

[1] becomes :
s = -4.9t² + C1t ...... [2]

From [0] :
C1 = (ds/dt) + 9.8t ...... [3]

Put [3] into [2] :
s = -4.9t² + [(ds/dt) + 9.8t]t
s = 4.9t² + (ds/dt)²t ...... [4]

Put s = 0 and ds/dt = 0 into [4] :
1 = 4.9t²
t = 1/√4.9

Put ds/dt = 0 and t = 1/√4.9 into [3] :
C1 = 0 + 9.8(1/√4.9)
C1 = √19.6
Then [2] becomes : s = -4.9t² + (√19.6)t ...... [5]

Put s = -10 into [5] :
-10 = -4.9t² + (√19.6)t
4.9t² - (√19.6)t - 10 = 0
t = [√19.6 ± √(19.6 + 4*4.9*10)] / (2*4.9)
t = 1.95 or -1.05 (rejected)

Hence, when s = -10, t = 1.95 s

"(2) When s = 1, ds/dt = 0" ?

Do you mean t = 1?