Question

The velocity (v) of a particle (under a force 'F') depends on its distance (x) from the origin (with x > 0). Find how the magnitude of the force (F) on the particle depends on x?

Answer 1

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:

1. The work done by the force is equal to the change in the particle's kinetic energy.
2. Calculate the work by integrating the force F(x) over the position x.
3. Determine the final velocity using the work-energy theorem.