Problem 18 - University Physics with Modern Physics

Question

(a) In the Bohr model of the atom, the ground state electron in hydrogen has an orbital speed of 2190 km/s. What is its kinetic energy? (Consult Appendix F.) (b) If you drop a 1.0-kg weight (about 2 lb) from a height of 1.0 m, how many joules of kinetic energy will it have when it reaches the ground? (c) Is it reasonable that a 30-kg child could run fast enough to have 100 J of kinetic energy?

Step-by-Step Solution

Answer

(a) KE = 13.6 eV; (b) KE = 9.8 J; (c) Yes, it's reasonable.

Step 1: Understand the Problem and Formulas Involved

For this problem, we need to calculate kinetic energy for three different situations. Kinetic energy (KE) is given by the formula \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object.

Step 2: Calculate Kinetic Energy of Ground State Electron

Given: velocity \( v = 2190 \text{ km/s} = 2.19 \times 10^6 \text{ m/s} \) (conversion from km/s to m/s) and mass of electron \( m = 9.109 \times 10^{-31} \text{ kg} \). Substituting in the formula:\[KE = \frac{1}{2} \times 9.109 \times 10^{-31} \text{ kg} \times (2.19 \times 10^6 \text{ m/s})^2\]Calculate the result and convert it to electron volts (eV) if necessary. Resulting KE: approximately \( 13.6 \text{ eV} \).

Step 3: Calculate Kinetic Energy of Dropped Weight

Given: mass \( m = 1.0 \text{ kg} \), height \( h = 1.0 \text{ m} \). The speed \( v \) just before hitting the ground, using \( v^2 = 2gh \), where \( g = 9.8 \text{ m/s}^2 \):\[ v^2 = 2 \times 9.8 \times 1.0 \rightarrow v = \sqrt{19.6} \approx 4.43 \text{ m/s} \]The kinetic energy is:\[KE = \frac{1}{2} \times 1.0 \times 4.43^2 \approx 9.8 \text{ J}\]

Step 4: Determine Reasonableness for a Child's Energy

Given: mass \( m = 30 \text{ kg} \). We need to check if \( KE = 100 \text{ J} \) is reasonable:\[KE = \frac{1}{2} \times 30 \times v^2 \rightarrow 100 = 15v^2 \rightarrow v^2 = \frac{100}{15} \approx 6.67 \rightarrow v \approx 2.58 \text{ m/s}\]At a velocity of \( 2.58 \text{ m/s} \), it's reasonable for a 30-kg child to have 100 J of kinetic energy.

Key Concepts

Bohr Model

In the early 20th century, Niels Bohr developed a simple yet revolutionary model of the atom to explain the behavior of electrons, especially in hydrogen atoms. The Bohr Model describes electrons as particles moving in fixed circular orbits around the nucleus. Each orbit corresponds to a specific energy level. This model helps to understand how electrons transition between energy levels by absorbing or emitting photons.
In terms of kinetic energy, the electron in its ground state possesses a specific velocity. The Bohr Model allows the calculation of this kinetic energy using the velocity and mass of the electron. Although modern quantum physics provides more accurate models, the Bohr Model remains a critical pedagogical tool for teaching basic atomic structure.

Electron Mass

Electrons are subatomic particles with a very small mass, critical for calculating various energy values in physics. The mass of an electron is approximately \(9.109 \times 10^{-31} \text{ kg}\), which is tiny compared to other atomic particles like protons or neutrons.
Knowing the electron's mass is essential, especially when you are dealing with scenarios like the kinetic energy of an electron in the Bohr Model. When given the velocity of an electron, its mass allows us to compute kinetic energy using the equation \( KE = \frac{1}{2}mv^2 \). It’s fascinating that even such a tiny mass can have measurable energy when moving at high velocities close to the speed of light.

Gravitational Potential Energy

Gravitational Potential Energy (GPE) represents the energy an object possesses due to its position in a gravitational field. The formula for GPE is \(U = mgh\), where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is the height above the ground.
In practical terms, when you drop an object from a height, it translates potential energy into kinetic energy as it falls. For instance, a 1 kg object dropped from a 1-meter height will convert its gravitational potential energy (\(9.8 \text{ J}\)) into kinetic energy just before it hits the ground. This transformation exemplifies the conservation of energy principle, showing how potential energy becomes active motion.

Velocity Calculation

Velocity calculation is crucial in physics to determine the speed and direction of an object's motion. When calculating kinetic energy, the velocity is squared, which means even small changes in velocity can significantly affect the energy, as shown in the equation \(KE = \frac{1}{2}mv^2\).
For example, using the equation \(v^2 = 2gh\), we calculated the velocity of a 1 kg object dropped from a 1-meter height to have approximately \(4.43 \text{ m/s}\). Another case is calculating if a running child with mass \(30 \text{ kg}\) reaches kinetic energy of \(100 \text{ J}\). Solving for velocity shows a speed of \(2.58 \text{ m/s}\). Therefore, understanding velocity calculation is vital to deducing how motion translates to kinetic energy.