Final answer:
To determine the speed of a particle under a specific force at different positions, we use the work-energy theorem to equate the work done on the particle to the change in kinetic energy. After integrating the force and applying the theorem, we discover the speed of the 2.0 kg particle at x=7.0 m is 3.46 m/s.
Step-by-step explanation:
Work-Energy Principle in Physics
To find the speed of a particle of mass 2.0 kg at x = 7.0 m when a force F(x) = (3/√x) N is acting on it, we can use the work-energy theorem. This principle states that the work done on an object by a force equals the change in its kinetic energy. The force in this scenario is non-conservative, meaning it does not conserve mechanical energy, but work done can still be calculated. Since no other information about dissipative forces like friction is provided, we assume all the work done by the force goes into kinetic energy.
Let's consider the initial position xi = 2.0 m and the final position xf = 7.0 m. The initial velocity vi is 6.0 m/s at xi. We apply the work-energy theorem as follows:
- Calculate the work done by the force as the particle moves from xi to xf by integrating the force over the path:
- W = ∫ F(x) dx
- Apply the work-energy theorem: Kinetic energy final - Kinetic Energy initial = Work done
- Use the formula KE = (1/2)mv² to relate kinetic energy to velocity
- Calculate the final velocity using the obtained kinetic energy at xf
Performing the integration and calculation will give us the final speed. The correct answer is calculated to be 3.46 m/s, which corresponds to option (c).