Answer:
A. V = 15 m/s
Explanation:
Equation to Use:
[tex]ME = KE + PE[/tex]
Total mechanical energy the sum of potential energy and kinetic energy.
Our potential energy is given as a function of position x:
[tex]U(x) = 8.0x^2 + 4.5x^4[/tex]
To determine the velocity of the particle at the origin, we need to obtain the total mechanical energy first.
Our knowns are:
[tex]m = 0.2 kg \\v(1) = 10 m/s[/tex]
Using our mechanical energy equation:
[tex]ME = 8.0x^2 + 4.5x^4 + \frac{1}{2}mv^2\\[/tex]
Substitute our values in.
[tex]ME = 8.0(1^2) + 4.5(1^4) + \frac{1}{2}(0.2)(10^2) \\ME = 8 + 4.5 + 10 \\ME = 22.5 J[/tex]
Now that we have our total mechanical energy, we can solve for the velocity using KE.
Taking into consideration that potential energy is based on position of the particle, the position of the particle at the origin is x = 0, which means that the potential energy at that point is zero. This indicates that the total mechanical energy is equivalent to the kinetic energy at that position.
[tex]ME = \frac{1}{2} mv^2 \\22.5J = \frac{1}{2}(0.2)v^2[/tex]
Solve for the velocity.
[tex]\frac{22.5(2)}{0.2} = v^2 \\225 = v^2 \\\sqrt{225} = v \\v = 15 m/s[/tex]
Thus, the velocity of the particle at the origin is 15 m/s.