An electron (m = 9.11 x 10⁻³¹ kg) is accelerated from rest to speed v by a conservative force. In this process, its potential energy decreases by 7.20 x 10⁻¹⁴ J . Determine the electron’s speed, v.

Verified step by step guidance

1. Identify the change in potential energy (\(\Delta U\)) and the mass of the electron (m). In this problem, \(\Delta U = -7.20 \times 10^{-14} \, J\) and \(m = 9.11 \times 10^{-31} \, kg\).
2. Understand that the decrease in potential energy implies an increase in kinetic energy by the same amount due to the conservation of energy. Thus, the change in kinetic energy (\(\Delta K\)) is equal to \(-\Delta U\).
3. Write the expression for kinetic energy, which is \(K = \frac{1}{2} m v^2\), where v is the speed of the electron.
4. Set up the equation \(\Delta K = \frac{1}{2} m v^2\) and substitute \(\Delta K\) with \(-\Delta U\) to find \(\frac{1}{2} m v^2 = -\Delta U\).
5. Solve for v by rearranging the equation to \(v = \sqrt{\frac{-2 \Delta U}{m}}\). Substitute the values of \(\Delta U\) and m to calculate the speed of the electron.

Key Concepts

Conservative Forces

Conservative forces are forces that do work on an object in such a way that the total mechanical energy (kinetic plus potential energy) of the system remains constant. The work done by a conservative force depends only on the initial and final positions of the object, not on the path taken. In this context, the conservative force acting on the electron is responsible for converting potential energy into kinetic energy as the electron accelerates.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v², where m is the mass and v is the velocity of the object. As the electron accelerates, the decrease in its potential energy is converted into kinetic energy, allowing us to determine its final speed. Understanding this relationship is crucial for solving the problem.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the decrease in potential energy of the electron is equal to the increase in its kinetic energy. By applying this principle, we can set the change in potential energy equal to the kinetic energy gained, allowing us to solve for the electron's speed.