Total energy of the system is "E = \\frac{k a^2}{2}", where "k" is the spring constant, and "a" is the amplitude. This quantity is conserved, and is equal to the sum of kinetic and potential energies at any moment: "E = \\frac{m v^2}{2} + \\frac{k x^2}{2}".
For given displacement and velocity, the equation "E = \\frac{m v_1^2}{2} + \\frac{k x_1^2}{2}" holds.
The object reaches its maximum speed when "x = 0", then "E = \\frac{m v_{max}^2}{2}", from where "v_{max} = \\sqrt \\frac{2 E}{m}".
Since we know the total energy in terms of amplitude "E = \\frac{k a^2}{2}" , let us express the mass from the equation of energy for given position and velocity "m = \\frac{ 2 E - k x_1^2}{v_1^2}"
, and substitute it into the expression for maximum velocity: "v_{max} = v_1 \\sqrt{\\frac{2 E}{2 E - k x_1^2}}" . Expressing energy in terms of the amplitude, and simplifying, obtain "v_{max} = v_1 \\sqrt{\\frac{a^2}{a^2 - x_1^2}}" .
Substituting "v_1 = 0.12 \\frac{m}{s}", "a = 0.2 m", "x_1 = 0.15 m", and calculating, obtain "v_{max} \\approx 0.18 \\frac{m}{s}".
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