Use Newtonian dynamics to find the speed at this position

[Jessikalinphy]

I need some help with this hwk! Please someone help me...
A certain spring is found not to obey Hook’s law; it exerts a restoring force if it is stretched or compressed, where α = 60.0 N / m and β = 18.0 N / m2. The mass of the spring is negligible. (a) Calculate the potential-energy function U(x) for the spring. Let U = 0 when x = 0. (b) An object with mass m = 0.900 kg on a frictionless, horizontal surface is attached to this spring, pulled a distance 1.00 m to the right (the + x-direction) to stretch the spring, and released. What is the speed of the object when it is 0.500 m to the right of the equilibrium position? (c) Use Newtonian dynamics to find the speed at this position. (d) What is the instantaneous power when x = 0.500 m?

 

[andrevdh]

According to the given coefficients it seems the relationship is of the form

$$F = \alpha x + \beta x^2$$

therefore

(a) use

$$U(x) = \int {F\ dx} + C$$

(b) use conservation of energy

(c) I am not sure what to do here maybe

$$a = \frac{dv}{dt} = \frac{F}{m}$$

therefore

$$\frac{dv}{dx} \frac{dx}{dt} = \frac{dv}{dx}v = \frac{F}{m}$$

giving

$$\frac{1}{2}v^2 = \int {\frac{F}{m}dx} + C$$

which in essense is conservation of energy again ?

 

[Jessikalinphy]

thankyou...