Two masses of \(3.4\) kg and \(2.5\) kg are accelerated from an initial speed of \(5\) m/s and \(12\) m/s, respectively. The distances traversed by the masses in the \(5^\text{th}\) second are \(104\) m and \(129\) m, respectively. The ratio of their momenta after \(10\) s is \(\dfrac{x}{8}\). The value of \(x\) is ________.

The distance traversed in the \(n^{\text{th}}\) second is given by \(S_n = u + \dfrac{a}{2}(2n - 1)\).

For the first mass (\(m_1 = 3.4\) kg, \(u_1 = 5\) m/s):
\(104 = 5 + \dfrac{a_1}{2}(2 \times 5 - 1)\)
\(99 = \dfrac{9a_1}{2} \Rightarrow a_1 = 22\) m/s\(^2\)

For the second mass (\(m_2 = 2.5\) kg, \(u_2 = 12\) m/s):
\(129 = 12 + \dfrac{a_2}{2}(2 \times 5 - 1)\)
\(117 = \dfrac{9a_2}{2} \Rightarrow a_2 = 26\) m/s\(^2\)

Velocities of the masses after \(10\) s:
\(v_1 = u_1 + a_1 t = 5 + 22 \times 10 = 225\) m/s
\(v_2 = u_2 + a_2 t = 12 + 26 \times 10 = 272\) m/s

Momenta of the masses after \(10\) s:
\(p_1 = m_1 v_1 = 3.4 \times 225 = 765\) kg m/s
\(p_2 = m_2 v_2 = 2.5 \times 272 = 680\) kg m/s

Ratio of their momenta:
\(\dfrac{p_1}{p_2} = \dfrac{765}{680} = \dfrac{9}{8}\)

Comparing with \(\dfrac{x}{8}\), we get \(x = 9\).

Answer: \(9\)