Question

The acceleration of a rocket traveling upward (Figure 1) is given by a=(6+0.02s)ms/s2, where s is in meters. Initially, v = 0 and s = 0, when t = 0. Determine the time when s = 2.5km

Answer 1

the differential equation is: s'' = 6 + 0.02s.

are you sure the problem is as presented here (no figure attached, BTW? the solution as the problem currently stands is best solved using, say, a Laplace Transform. is that what you expect?

that is well in advance of what usually appears in these parts. but happy to oblige, if so.

in meantime, building a numerical solution in a spreadsheet or using code is the way this would be solved in the real world.

Answer 2

the exact (LaPlace) solution incidentally is:
s(t) = 150 { exp(ßt) + exp(-ßt) - 2 }

where ß = sqrt (0.02)

this gives t ~ 20.7 sec before s = 2500m