Hello,

There are several equations need to be solve together.

1. v'[t] == k*s*Sqrt[2*9.8*h]
2. r == Sqrt[1 - (1 - h)^2]
3. v'[h] == -Pi*r^2
4. k == 0.62
5. s == 10^-4
6. When t equals 0, h(t)==1

So, I make out code as

DSolve[{D[v[t, h], t] == k*s*Sqrt[2*9.8*h], r == Sqrt[1 - (1 - h)^2], 
  D[v[t, h], h] == -Pi*r^2, k == 0.62, s == 10^-4, h[0] == 1}, v, {t, 
  h}]

I know this is not correct, at least h[0]==1 is not correct. How to find the relationship between t and h?

This one can work but not give correct answer

DSolve[{D[v[t, h], t] == k*s*Sqrt[2*9.8*h], 
  D[v[t, h], h] == -Pi*r^2}, v, {t, h}, 
 Assumptions -> {r == Sqrt[1 - (1 - h)^2], k == 0.62, s == 10^-4, 
   h == 1 /; t == 0}]

The answer should be t == 1.068*10^4*(1 - 10/7*h^(3/2) + 3/7*h (5/2))

Thanks.