91Ó°ÊÓ

Kinetic energy is the energy an object possesses due to its motion. The formula to calculate kinetic energy is given by: \( K = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. In our exercise, an electron is projected toward a proton. Initially, the electron has a kinetic energy based on its speed of \( 3.2 \times 10^5 \) m/s. As it moves and accelerates, its velocity increases, thus changing its kinetic energy. Ultimately, understanding kinetic energy allows us to set up equations involving energy changes due to motion.

Potential energy is the energy stored within a system due to the positions of its components. For an electron and a proton, we consider electrostatic potential energy, given by: \( U = -\frac{k e^2}{r} \). Here, \( k \) is Coulomb's constant, \( e \) is the electron charge, and \( r \) is the distance between the electron and the proton. In our problem, initially, the potential energy is zero because the electron is far from the proton. As the electron moves closer, it gains potential energy, which impacts the overall energy conservation equation we use to find the distance at which the kinetic energy parameters change.

The electrostatic force is the force acting between charged particles due to their electric charges. Coulomb's law describes this force: \( F = k \frac{e_1 e_2}{r^2} \), where \( e_1 \) and \( e_2 \) are the charges and \( r \) is the distance between them. For our electron and proton, this force attracts the electron towards the proton. This electrostatic attraction changes the electron's speed and direction, which is why it reaches a velocity of twice its initial value at a certain point. Understanding this force helps explain the energy changes described in conservation of energy principles.

The principle of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another. In our exercise, the total mechanical energy of the electron-proton system (kinetic + potential energy) remains constant. Initially, the electron's total energy is entirely kinetic due to its motion, as it's far from the proton. As the electron moves closer, its kinetic energy increases while its potential energy decreases. By setting up the conservation of energy equation, we balance the initial and final energies to find the distance where the electron's speed is twice its initial value. This critical insight helps solve for unknown variables such as distance in various physics problems.