a 30.0 g particle is released from rest at x = 5 J. What is the speed of the particle at x = 0 J?

To determine the speed of the particle at x = 0 J, we can apply the principles of conservation of mechanical energy.

The mechanical energy of the particle is given by the sum of its kinetic energy (KE) and potential energy (PE). At any point, the total mechanical energy remains constant.

Given that the particle is released from rest (initially at x = 5 J), there is no initial kinetic energy. Hence, the initial mechanical energy is equal to the potential energy at x = 5 J.

We can calculate the initial potential energy (PEi) using the formula:

PEi = mgh

Where:
m = mass of the particle (30.0 g = 0.03 kg)
g = acceleration due to gravity (approximately 9.8 m/s^2 on Earth)
h = height or displacement from the reference point (given as x = 5 J)

PEi = (0.03 kg)(9.8 m/s^2)(5 m)

Next, we can determine the final potential energy (PEf) at x = 0 J. Since the particle is at x = 0 J, the height or displacement from the reference point is zero, resulting in no potential energy.

PEf = 0 J

Since mechanical energy is conserved, we can equate the initial mechanical energy (PEi) to the final mechanical energy (KEf + PEf):

PEi = KEf + PEf

Since PEi = PEf (zero potential energy) and KEf is the final kinetic energy, we have:

0 = KEf + 0

Thus, the final kinetic energy (KEf) is also zero. This implies that the speed of the particle at x = 0 J is zero, as kinetic energy is directly related to the square of the speed (KE = 0.5 * m * v^2).