Answer:

7.85 m/s

Explanation:

We are given that

Mass of object=m=0.900 kg

[tex]F(x)=\alpha x-\beta x^2[/tex]

[tex]\alpha=60 N/m[/tex]

[tex]\beta=18N/m^2[/tex]

[tex]F(x)=-60x-18x^2[/tex]

U=0 when x=0

Potential energy=[tex]-\int F(x)dx[/tex]

Substitute the values

[tex]U(x)=-\int (-60x-18x^2)dx[/tex]

[tex]U(x)=60(\frac{x^2}{2})+18(\frac{x^3}{3})+C[/tex]

Using the formula

[tex]\int x^n dx=\frac{x^{n+1}}{n+1}+C[/tex]

Substitute x=0

[tex]U(0)=C\implies C=0[/tex]

[tex]U(x)=30x^2+6x^3[/tex]

[tex]x_1=0.5,x_2=1[/tex]

[tex]v_2=0[/tex]

Using law of conservation energy

[tex]\frac{1}{2}mv^2_1+U(x_1)=\frac{1}{2}mv^2_2+U(x_2)[/tex]

Substitute the values

[tex]\frac{1}{2}(0.9)v^2_1+30(0.5)^2+6(0.5)^3=0+30(1)^2+6(1)^3[/tex]

[tex]\frac{1}{2}(0.9)v^2_1+8.25=36[/tex]

[tex]\frac{1}{2}(0.9)v^2_1=36-8.25=27.75[/tex]

[tex]v^2_1=\frac{27.75\times 2}{0.9}[/tex]

[tex]v_1=\sqrt{\frac{27.75\times 2}{0.9}}[/tex]

[tex]v_1=7.85 m/s[/tex]