a) \(\displaystyle{\frac{{{1}}}{{{2}}}}{k}{x}^{{2}}={\frac{{{1}}}{{{2}}}}{m}{v}^{{2}}\)

\(\displaystyle{v}=\sqrt{{{\frac{{{k}{x}^{{2}}}}{{{m}}}}}}=\sqrt{{{\frac{{{\left({400}\right)}{\left({0.06}\right)}^{{2}}}}{{{0.03}}}}}}={6.93}\ {\frac{{{m}}}{{{s}}}}\)

b) \(\displaystyle{\frac{{{1}}}{{{2}}}}{m}{v}^{{2}}={W}_{{\text{total}}}\)

\(\displaystyle{\frac{{{1}}}{{{2}}}}{m}{v}^{{2}}={\frac{{{1}}}{{{2}}}}{k}{x}^{{2}}-{F}_{{{\mathfrak{{i}}}{c}}}{x}\)

\(\displaystyle{v}=\sqrt{{{\frac{{{k}{x}^{{2}}-{2}{F}_{{{\mathfrak{{i}}}{c}}}{x}}}{{{m}}}}}}\)

\(\displaystyle{v}=\sqrt{{{\frac{{{\left({400}\right)}{\left({0.06}\right)}^{{2}}-{2}{\left({6}\right)}{\left({0.06}\right)}}}{{{0.03}}}}}}={4.90}\ {\frac{{{m}}}{{{s}}}}\)

c) The greatest speed occurs when the acceleration (and the net force)are zero

\(\displaystyle{F}={k}{x}\)

\(\displaystyle{x}={\frac{{{F}}}{{{k}}}}={\frac{{{6}}}{{{400}}}}={0.015}{m}\)

Then to find the speed:

\(\displaystyle{W}_{{\text{total}}}={\frac{{{1}}}{{{2}}}}{k}{\left({{x}_{{0}}^{{2}}}-{x}^{{2}}\right)}-{f{{\left({x}_{{0}}-{x}\right)}}}\)

Solve for velocity;

\(\displaystyle{v}={5.20}\ {\frac{{{m}}}{{{s}}}}\)

\(\displaystyle{v}=\sqrt{{{\frac{{{k}{x}^{{2}}}}{{{m}}}}}}=\sqrt{{{\frac{{{\left({400}\right)}{\left({0.06}\right)}^{{2}}}}{{{0.03}}}}}}={6.93}\ {\frac{{{m}}}{{{s}}}}\)

b) \(\displaystyle{\frac{{{1}}}{{{2}}}}{m}{v}^{{2}}={W}_{{\text{total}}}\)

\(\displaystyle{\frac{{{1}}}{{{2}}}}{m}{v}^{{2}}={\frac{{{1}}}{{{2}}}}{k}{x}^{{2}}-{F}_{{{\mathfrak{{i}}}{c}}}{x}\)

\(\displaystyle{v}=\sqrt{{{\frac{{{k}{x}^{{2}}-{2}{F}_{{{\mathfrak{{i}}}{c}}}{x}}}{{{m}}}}}}\)

\(\displaystyle{v}=\sqrt{{{\frac{{{\left({400}\right)}{\left({0.06}\right)}^{{2}}-{2}{\left({6}\right)}{\left({0.06}\right)}}}{{{0.03}}}}}}={4.90}\ {\frac{{{m}}}{{{s}}}}\)

c) The greatest speed occurs when the acceleration (and the net force)are zero

\(\displaystyle{F}={k}{x}\)

\(\displaystyle{x}={\frac{{{F}}}{{{k}}}}={\frac{{{6}}}{{{400}}}}={0.015}{m}\)

Then to find the speed:

\(\displaystyle{W}_{{\text{total}}}={\frac{{{1}}}{{{2}}}}{k}{\left({{x}_{{0}}^{{2}}}-{x}^{{2}}\right)}-{f{{\left({x}_{{0}}-{x}\right)}}}\)

Solve for velocity;

\(\displaystyle{v}={5.20}\ {\frac{{{m}}}{{{s}}}}\)