ClearAll["Global`*"]
q1 = NDSolve[{q'[s] ==
Sqrt[q[s]/Sqrt[u[s]^2 + 1]]*(abs (u[s]))/Sqrt[
 u[s]^2 + 1]*(abs (u[s]) + 5/Sqrt[u[s]^2 + 1]),
 q[s]*u'[s] + q'[s]*u[s] == -(q[s]/Sqrt[u[s]^2 + 1]), q[0] == 0.025,
u[0] == 3.24}, {q, u}, {s, 0, 5}]
Plot[{{u[s] /. q1}, {q[s] /. q1}}, {s, 0, 5}]