Final answer:
The question involves using the work-energy theorem to find the speed of a 2.0 kg particle at a different position, given the force acting on it and its initial speed.
Step-by-step explanation:
The question involves determining the speed of a particle of mass 2.0 kg at position x = 4.0 m, given that it moves under the influence of a force F(x) = (-5x² + 7x) N and that its speed at x = -4.0 m is 20.0 m/s. To find the speed at x = 4.0 m, we can use the work-energy theorem, which relates the work done on a particle to its change in kinetic energy. Since the force is conservative, we can use the potential energy function associated with the force to find the change in kinetic energy as the particle moves from x = -4.0 m to x = 4.0 m.
The potential energy function U(x) is obtained by integrating the force with respect to x, leading to U(x) = ∫ F(x) dx. The difference in potential energy between the two points gives us the work done on the particle, which equals the change in kinetic energy since there are no non-conservative forces at work. We can then solve for the final speed using the relation K.E. = ½ mv².
It's important to note that kinetic energy is always positive, as it is proportional to the square of the speed, and mass. Therefore, this problem involves conserving mechanical energy and applying principles of classical mechanics.