\[Пусть\ \text{x\ }\frac{км}{ч} - скорость\ первого\ катера;\]
\[\text{y\ }\frac{км}{ч} - скорость\ второго\ катера.\]
\[Составим\ систему\ уравнений:\ \]
\[\left\{ \begin{matrix} x + y = 50\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \frac{60}{x} - \frac{60}{y} = 1\ \ \ | \cdot xy \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x = 50 - y\ \ \ \ \ \ \ \ \ \ \\ 60y - 60x = xy \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x = 50 - y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 60y - 60(50 - y) = y(50 - y) \\ \end{matrix} \right.\ \]
\[60y - 3000 + 60y = 50y - y^{2}\]
\[y^{2} + 120y - 50y - 3000 = 0\]
\[y^{2} + 70y - 3000 = 0\]
\[D = 1225 + 3000 = 4225 = 65^{2}\]
\[y_{1} = - 35 + 65 = 30;\]
\[y_{2} = - 35 - 65 = - 100\ (не\ подходит).\]
\[\left\{ \begin{matrix} y = 30 \\ x = 20 \\ \end{matrix} \right.\ \]
\[Ответ:20\ \frac{км}{ч}\ и\ 30\ \frac{км}{ч}\text{.\ }\]