Question

A gold nucleus of rest mass \(183.473 \mathrm{GeV} / \mathrm{c}^{2}\) is accelerated from \(0.5785 c\) to \(0.8433 c .\) How much work is done on the gold nucleus in this process?

Short answer

Question: Calculate the work done on a gold nucleus, with a rest mass of 3.27 x 10^{-25} kg, during an acceleration process that changes its velocity from 0.5785c to 0.8433c. Step 1: Calculate the initial relativistic gamma factor. \(\gamma_{i} = \frac{1}{\sqrt{1 - (v_{i}/c)^2}}\) \(\gamma_{i} = \frac{1}{\sqrt{1 - (0.5785)^2}} \approx 1.332\) Step 2: Calculate the final relativistic gamma factor. \(\gamma_{f} = \frac{1}{\sqrt{1 - (v_{f}/c)^2}}\) \(\gamma_{f} = \frac{1}{\sqrt{1 - (0.8433)^2}} \approx 1.900\) Step 3: Calculate the initial and final kinetic energies. \(K_{i} = (\gamma_{i} - 1)mc^2 \approx (1.332 - 1)(3.27 × 10^{-25} kg)(3 × 10^8 m/s)^2 \approx 1.47 × 10^{-11} J\) \(K_{f} = (\gamma_{f} - 1)mc^2 \approx (1.900 - 1)(3.27 × 10^{-25} kg)(3 × 10^8 m/s)^2 \approx 2.65 × 10^{-11} J\) Step 4: Calculate the work done. \(W = K_{f} - K_{i} = 2.65 × 10^{-11} J - 1.47 × 10^{-11} J \approx 1.18 × 10^{-11} J\) Therefore, the work done on the gold nucleus during the acceleration process is approximately 1.18 × 10^{-11} Joules.

Step-by-step solution

Calculate the initial relativistic kinetic energy.

For a particle moving at relativistic speeds, the relativistic kinetic energy is given by: \(K_{i} = (\gamma_{i} - 1)mc^2\) where \(K_{i}\) is the initial kinetic energy, \(\gamma_{i}\) is the relativistic gamma factor for the initial velocity, \(m\) is the rest mass of the gold nucleus, and \(c\) is the speed of light. First, we need to calculate the initial gamma factor: \(\gamma_{i} = \frac{1}{\sqrt{1 - (v_{i}/c)^2}}\) where \(v_{i} = 0.5785 c\) is the initial velocity of the gold nucleus.

Calculate the final relativistic kinetic energy.

The final relativistic kinetic energy can be calculated in the same way as the initial kinetic energy: \(K_{f} = (\gamma_{f} - 1)mc^2\) where \(K_{f}\) is the final kinetic energy and \(\gamma_{f}\) is the relativistic gamma factor for the final velocity. The final gamma factor can be calculated using the same formula as before: \(\gamma_{f} = \frac{1}{\sqrt{1 - (v_{f}/c)^2}}\) where \(v_{f} = 0.8433 c\) is the final velocity of the gold nucleus.

Calculate the work done.

Now that we have the initial and final kinetic energies, we can calculate the work done by using the work-energy theorem: \(W = K_{f} - K_{i}\) This will give us the work done on the gold nucleus during the acceleration process.

Final calculation.

Plug in the initial and final velocities, rest mass, and speed of light into the formulas for the initial and final gamma factors, kinetic energies, and then calculate the work done using the work-energy theorem. This will give you the work done on the gold nucleus during the acceleration process.

Key concepts

Relativity in Physics

Relativity in physics refers to the two theories established by Albert Einstein: special relativity and general relativity. Special relativity is especially important when dealing with objects moving at speeds close to the speed of light, denoted by the symbol c. This theory introduced revolutionary concepts such as the idea that the laws of physics are the same for all non-accelerating observers, and that the speed of light within a vacuum is constant, regardless of the motion of the light source or observer.

One radical outcome of special relativity is that time can slow down, and lengths can contract for objects in motion relative to an observer—a phenomenon known as time dilation and length contraction. Additionally, relativity affects the mass-energy equivalence, best known by the formula \( E=mc^2 \), and has significant implications for understanding the relativistic kinetic energy of particles moving at significant fractions of the speed of light. These effects become noticeable as the velocity of an object approaches the speed of light, as the gold nucleus in the original exercise experiences. In such cases, classical formulas for kinetic energy cannot be used, requiring a relativistic approach instead.

Work-Energy Theorem

The work-energy theorem is a fundamental principle in physics that states that the work done by all forces acting on a particle equals the change in its kinetic energy. Formally, it is expressed as \( W = \Delta K \), where \( W \) represents work and \( \Delta K \) is the change in kinetic energy.

Applying this theorem in relativistic contexts, we consider the relativistic kinetic energy of a particle, not the classical one. In the case of the gold nucleus being accelerated, the work done is the difference between the nucleus's final kinetic energy and its initial kinetic energy. The relativistic kinetic energy of an object exceeds the classical expectation, and the disparity grows larger as the object's velocity increases. This principle explains why an enormous amount of energy is needed to accelerate an object as it nears the speed of light, as the work done contributes to both the kinetic energy and the mass energy of the object because of the mass-energy equivalence.

Gamma Factor Calculation

The gamma factor is a dimensionless quantity that arises in the Lorentz transformation equations of special relativity. It is denoted by \( \gamma \) and provides a measure of time dilation, length contraction, and relativistic mass. The gamma factor is calculated using the formula \( \gamma = \frac{1}{{\sqrt{1 - (v/c)^2}}} \), where \( v \) is the velocity of the object and \( c \) is the speed of light.

The larger the object's velocity \( v \) relative to the speed of light, the greater the gamma factor becomes, indicating a stronger relativistic effect. In the context of the original exercise, the gold nucleus's initial and final velocities require different gamma factors to determine the relativistic kinetic energies at those speeds. For the gold nucleus starting at \( 0.5785c \) and accelerating to \( 0.8433c \), calculating the gamma factors for both velocities is crucial because it gives us the multiplier needed to compute the kinetic energy correctly under relativistic conditions. Indeed, as the velocity increases, the gamma factor increases, illustrating how even a small change in velocity at such high speeds can result in a significant amount of work and energy required for acceleration.