Final answer:
The mechanical energy of the particle can be calculated by finding its potential energy at each reference point. The velocity of the particle at x = 1.0 m can be found using the conservation of mechanical energy.
Step-by-step explanation:
To calculate the mechanical energy of the particle, we need to find its potential energy at each reference point. The potential energy at a given point is given by the negative integral of the force from the reference point to that point. So, for (a) the origin as the reference point, the potential energy at x = 2.0 m is:
Potential energy = - ∫F(x) dx from 0 to 2.0 = - ∫(-3.0x²) dx from 0 to 2.0 = - (-x³) from 0 to 2.0 = -(-8) = 8 J
For (b) x = 4.0 m as the reference point, the potential energy at x = 2.0 m is:
Potential energy = - ∫F(x) dx from 4.0 to 2.0 = - ∫(-3.0x²) dx from 4.0 to 2.0 = - (-x³) from 4.0 to 2.0 = -(-64/3 + 8) = 64/3 - 8 ≈ 8.13 J
To find the particle's velocity at x = 1.0 m, we need to use the conservation of mechanical energy. The mechanical energy of the particle is the sum of its kinetic energy and potential energy. Since the potential energy is calculated above, we can subtract it from the total mechanical energy to find the kinetic energy. Then, we can use the kinetic energy equation to find the velocity.
For (a) the origin as the reference point:
Mechanical energy = kinetic energy + potential energy = 0.5 * (5.0)² + 8 = 6.25 + 8 = 14.25 J
Kinetic energy = Mechanical energy - potential energy = 14.25 - 8 = 6.25 J
Velocity = √(2 * kinetic energy / mass) = √(2 * 6.25 / 2) = √6.25 = 2.5 m/s
For (b) x = 4.0 m as the reference point:
Mechanical energy = kinetic energy + potential energy = 0.5 * (5.0)² + (64/3 - 8) ≈ 6.25 + 8.13 ≈ 14.38 J
Kinetic energy = Mechanical energy - potential energy = 14.38 - (64/3 - 8) ≈ 6.25 + 8.13 ≈ 6.38 J
Velocity = √(2 * kinetic energy / mass) = √(2 * 6.38 / 2) = √6.38 ≈ 2.52 m/s