Joshua,

Your problem is not formulated correctly. If you want to integrate your original problem use

NDSolve[{y''[t] == 5/y[t]^2, y[0] == 100, y'[0] == 0}, 
 y[t], {t, 0, 100}]

Your second version of the problem does not make sense. I can't understand how it follows from the first equation and besides, Its a first order ODE so you can only specify one initial condition (y[0]==0). You can't specify y'[0] because the differential equation is for y'[t]. To integrate the second equation (assuming it had some physical meaning) the proper formulation in MMA would be as follows (since y'[0] ==0 there can be no y'[0] term)

NDSolve[{y'[t]^2/2 == -5/y[t], y[0] == 100}, y[t], {t, 0, 100}]

I hope this helps.

Regards,

Neil