Final answer:
The magnitude of the force F on a particle depends on its distance x from the origin through the work-energy theorem, which relates the work done by the force to the change in kinetic energy of the particle.
Step-by-step explanation:
To determine how the magnitude of the force F on a particle depends on its distance x from the origin, we must consider the work-energy theorem. Given that a particle has a mass of 2.0 kg and moves under the influence of the force F(x) = (3/√√x) N, we can relate the work done by the force to the change in kinetic energy of the particle as it moves from one position to another.
From the work-energy principle, the work done on the particle equals the change in kinetic energy, which can be calculated by:
W = ∆KE = ½ m(v22 - v12),
where m is the mass of the particle, v1 is the initial velocity, and v2 is the final velocity.
For a force that is a function of x, as is the case here, work done (W) can also be calculated by integrating the force over the distance:
W = ∫ F(x) dx.
We can then set up the integral with the given force function to find the relationship between velocity and position.
The speed v at position x = 7.0 m can be found after doing these calculations, starting with the given speed of v = 6.0 m/s at x = 2.0 m.
SUMUP all the final answer as points at last:
- The work done by the force is equal to the change in the particle's kinetic energy.
- Calculate the work by integrating the force F(x) over the position x.
- Determine the final velocity using the work-energy theorem.