A $20\mathrm{kg}$ object is acted on by a conservative force given by $F=-3.0x-5.0x^2$, with $F$ in newtons and $x$ in meters. Take the potential energy associated with the force to be zero when the object is at $x=0.$ (a) What is the potential energy of the system associated with the force when the object is at $x=2.0\mathrm{m}?(\mathrm{b})$ If the object has a velocity of $4.0\mathrm{m}/\mathrm{s}$ in the negative direction of the $x$ axis when it is at $x=5.0\mathrm{m}$, what is its speed when it passes through the origin? (c) What are the answers to (a) and (b) if the potential energy of the system is taken to be $-8.0\mathrm{J}$ when the object is at $x=0?$

Step by Step Solution: Step 1. Calculate the potential energy of the object at x=2.0m using the provided force equation. At x = 0, the potential energy is zero. Step 2. Calculate the initial potential energy of the system based on the provided information in the problem statement. Step 3. Use the Law of Conservation of Energy to solve for the final speed of the object at x=0. Final Answer: a) The potential energy of the system when the object is at x=2.0m is -30J. b) The speed of the object when it passes through the origin is 6.32m/s. c) a) The potential energy of the system when the object is at $x=2.0m$ is -38J. b) The speed of the object when it passes through the origin is 9.85m/s.