Problem 41 - University Physics with Modern Physics

Question

Compute the kinctic energy of a proton (mass \(1.67 \times$$10^{-27} \mathrm{kg}\) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) \(8.00 \times 10^{7} \mathrm{m} / \mathrm{s}\) and (b) \(2.85 \times 10^{8} \mathrm{m} / \mathrm{s}\) .

Step-by-Step Solution

Answer

For speed (a), the ratio is close to 1. For speed (b), the ratio is significantly greater than 1.

Step 1: Define Nonrelativistic Kinetic Energy

The formula for nonrelativistic kinetic energy is given by \( KE_{nr} = \frac{1}{2}mv^2 \). Here, we have the mass of the proton \( m = 1.67 \times 10^{-27} \; \text{kg} \) and the velocity \( v \). We will calculate \( KE_{nr} \) for both speeds.

Step 2: Calculate Nonrelativistic Kinetic Energy for Speed (a)

Substitute \( v = 8.00 \times 10^7 \; \text{m/s} \) into the nonrelativistic formula: \[KE_{nr, (a)} = \frac{1}{2} \times 1.67 \times 10^{-27} \times (8.00 \times 10^7)^2\] Calculate to find \( KE_{nr, (a)} \).

Step 3: Calculate Nonrelativistic Kinetic Energy for Speed (b)

Substitute \( v = 2.85 \times 10^8 \; \text{m/s} \) into the nonrelativistic formula: \[KE_{nr, (b)} = \frac{1}{2} \times 1.67 \times 10^{-27} \times (2.85 \times 10^8)^2\] Calculate to find \( KE_{nr, (b)} \).

Step 4: Define Relativistic Kinetic Energy

The formula for relativistic kinetic energy is \( KE_r = (\gamma - 1)mc^2 \), where \( \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \) and \( c = 3 \times 10^8 \; \text{m/s} \). We will calculate \( KE_r \) for both speeds.

Step 5: Calculate Relativistic Kinetic Energy for Speed (a)

First, find \( \gamma \) for \( v = 8.00 \times 10^7 \; \text{m/s} \):\[\gamma = \frac{1}{\sqrt{1-\left(\frac{8.00 \times 10^7}{3 \times 10^8}\right)^2}}\]Then, calculate \( KE_r \):\[KE_{r, (a)} = (\gamma - 1) \times 1.67 \times 10^{-27} \times (3 \times 10^8)^2\]

Step 6: Calculate Relativistic Kinetic Energy for Speed (b)

First, find \( \gamma \) for \( v = 2.85 \times 10^8 \; \text{m/s} \):\[\gamma = \frac{1}{\sqrt{1-\left(\frac{2.85 \times 10^8}{3 \times 10^8}\right)^2}}\]Then, calculate \( KE_r \):\[KE_{r, (b)} = (\gamma - 1) \times 1.67 \times 10^{-27} \times (3 \times 10^8)^2\]

Step 7: Calculate the Ratio of Relativistic to Nonrelativistic Kinetic Energy for Speed (a)

The ratio \( R_{(a)} \) is given by:\[R_{(a)} = \frac{KE_{r, (a)}}{KE_{nr, (a)}}\]Calculate \( R_{(a)} \) using the values obtained from the previous steps.

Step 8: Calculate the Ratio of Relativistic to Nonrelativistic Kinetic Energy for Speed (b)

The ratio \( R_{(b)} \) is given by:\[R_{(b)} = \frac{KE_{r, (b)}}{KE_{nr, (b)}}\]Calculate \( R_{(b)} \) using the values obtained from the previous steps.

Key Concepts

Nonrelativistic Kinetic Energy

The concept of nonrelativistic kinetic energy is tied closely to what we typically learn in basic physics about energy in motion. Simply put, kinetic energy is the energy an object has due to its motion. For slow-moving objects, especially those moving at much less than the speed of light, the classical or nonrelativistic formula for kinetic energy works well: \[ KE_{nr} = \frac{1}{2}mv^2 \] Here, \( m \) represents mass, and \( v \) is the velocity of the object. In our scenario, we're calculating the kinetic energy of a proton, whose mass is \( 1.67 \times 10^{-27} \mathrm{kg} \). This formula gives a good estimate of kinetic energy when the velocity is a small fraction of the speed of light, which is common in everyday situations.

Relativistic Kinetic Energy

As speeds approach that of light, Einstein’s theory of special relativity shows that classical physics doesn't quite hold up. The relativistic kinetic energy accounts for the increased mass of an object as it moves ever faster. The formula is:\[ KE_r = (\gamma - 1)mc^2 \] Here, \( c \) is the speed of light, and \( \gamma \) (gamma) is the Lorentz factor, calculated as:\[ \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \] This factor becomes significant at speeds close to the speed of light, reflecting the dependency of a particle’s energy on its velocity. This equation is crucial for understanding high-speed particles in systems like particle accelerators.

Proton Mass

When dealing with particles like protons, their mass is a foundational element in any kinetic energy calculation. The proton is one of the fundamental constituents of atoms, having a mass of approximately \( 1.67 \times 10^{-27} \mathrm{kg} \). Though seemingly negligible, this mass is crucial in scientific calculations, especially in physics experiments and theoretical calculations. Protons, being part of the atomic nucleus, interact under forces including the strong nuclear force, making their mass essential in both classical and modern physics calculations.

Speed of Light

In physics, the speed of light (\( c \), approximately \( 3 \times 10^8 \mathrm{m/s} \)) is a significant constant that plays an essential role in the structure of the universe. It is not just a measure of the fastest speed at which information or matter can travel, but a fundamental component of Einstein's mass-energy equivalence and special relativity. Understanding its role is key when considering relativistic effects, where velocities approaching this constant hugely influence the behavior and energy of moving objects, as seen in our relativistic energy formulas.

Special Relativity

The groundbreaking theory of special relativity was introduced by Albert Einstein in 1905. It revolutionized how we understand time, space, and energy. This theory demonstrates how the laws of physics are identical for all observers in uniform motion relative to one another, introducing the concept that the speed of light is constant in vacuum. It reshapes how we calculate energy and momentum for objects moving at high velocities, leading to a more accurate understanding of kinetic energy beyond classical Newtonian mechanics. This understanding allows scientists to predict the behavior of particles moving close to the speed of light, an essential aspect of modern physics.