5.1 The solution for Re < 0.5

  Clear[v,m,g,k]
  vsolution1 = 
       DSolve[{m v'[t] == -m g - k v[t],v[0] == v0}, v[t], t]

We need to select the right hand side of the above replacement rule.

  velocity1 = vsolution1[[1,1,2]]

Now integrate to get y[t].

  Clear[v,m,g,k]
  ysolution1 = 
       DSolve[{y'[t] == velocity1, y[0] == y0}, y[t], t]

  Clear[v,m,g,k]
  height1 = ysolution1[[1,1,2]]

Up to 5. Solutions of the Differential Equations