A uranium nucleus (mass 238 u, charge \(92 e\) ) decays, emitting an alpha particle (mass 4 u, charge \(2 e\) ) and leaving a thorium nucleus (mass 234 u, charge \(90 e\) ). At the instant the alpha particle leaves the nucleus, the centers of the two are 7.4 fm apart and essentially at rest. Treating each particle as a spherical charge distribution, find their speeds when they're a great distance apart.

When two charged particles are close together, they have a certain amount of potential energy due to their position. This type of energy is known as electrostatic potential energy. It's the energy stored due to the electrostatic forces between charges. In the case of nuclear decay, like the uranium nucleus decaying into an alpha particle and a thorium nucleus, the electrostatic potential energy can be significant due to the very strong electrical forces at play within such small distances.

For two point charges, the electrostatic potential energy (\( U \)) is calculated by the formula \( U = k \cdot \frac{q_1 \cdot q_2}{r} \), where \( k \) is the Coulomb constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance separating them. When the charges come from atoms as in our exercise, their charges are quantized in units of the elementary charge (\( e \) ), and distance must be converted from femtometers (fm) to meters (m) to use standard units.

Understanding this concept is vital in predicting the behavior of particles during nuclear reactions and interpreting such events in terms of energy transformations.

One of the fundamental principles of physics is the law of conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form to another. In the context of nuclear decay, this law implies that the total energy before and after the decay must be the same.

When the uranium nucleus in our exercise decays, the potential energy due to the electrostatic force between the nuclear fragments (the alpha particle and the thorium nucleus) is converted into kinetic energy as they move apart. Calculating these energy transformations allows us to determine the speed of the particles after decay, as all the initial potential energy will have become kinetic energy at the point when they are sufficiently far apart to neglect the nuclear forces.

Applying the conservation of energy principle is crucial in solving problems related to nuclear physics and understanding how energy is redistributed during such processes.

Kinetic energy is the energy that an object possesses due to its motion. It can be calculated using the equation \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. In nuclear decay, as in the uranium decay mentioned in our exercise, the alpha particle and the thorium nucleus gain kinetic energy as they move away from each other.

Once we calculate the initial electrostatic potential energy, we can use the law of conservation of energy to determine the kinetic energies of the alpha particle and the thorium nucleus. By rearranging the kinetic energy formula, we find the velocities of the particles: \( v = \sqrt{\frac{2 \cdot KE}{m}} \). This allows us to compute their final speeds assuming no other forces act upon them after a certain distance.

Understanding kinetic energy is crucial for analyzing motion in all areas of physics, from the smallest atomic particles to vast celestial bodies in motion.