Simple problem using Work-Energy Theorem

[mbrmbrg]

Take Two: problem using Work-Energy Theorem

The only force acting on a 1.5 kg body as it moves along the positive x-axis has an x component Fx = - 6x N, where x is in meters. The velocity of the body at x = 3.0 m is 8.0 m/s.
(a) What is the velocity of the body at x = 4.0 m?
(b) At what positive value of x will the body have a velocity of 5.0 m/s?

WebAssign says that this problem uses concepts from the sections on Work Done by a Spring Force and Work and Kinetic Energy. I did not see how spring concepts were relevant, so I ignored them, though when I evaluated the integral, it looked a lot like a spring.
I got part (b) correct, but I'm down to my last response for part (a). Can someone tell me if this looks all right?

$$W=\int{F(x)dx} = \int{-6xdx} = -6\frac{x^2}{2} = -3x^2$$
I evaluated the integral between x_i=3 and x_f=4 to get $$W = -3(4^2)-(-3)(3^2) = -21$$

$$W=\Delta K=\frac{m}{2}(v_f^2-v_i^2)$$
so $$\frac{2W}{m}+v_i^2=v_f^2$$
so $$v_f = \sqrt{36} = 6m/s$$

Thanks!

 

[SGT]

Your solution is correct. The equation F = -kx, where x is the displacement, is typical of a spring.

 

[mbrmbrg]

Thank you very much; getting a fairly simple problem wrong four times in a row does something to a person's self-confidence.