Question

A gold nucleus of rest mass \(183.473 \mathrm{GeV} / c^{2}\) is accelerated from \(0.4243 c\) to some final speed. In this process, \(140.779 \mathrm{GeV}\) of work is done on the gold nucleus. What is the final speed of the gold nucleus as a fraction of \(c ?\)

Short answer

Answer: The final speed of the gold nucleus is approximately 0.8833c.

Step-by-step solution

Apply the Work-Energy Theorem

The work-energy theorem states that the net work done on an object equals the change in its kinetic energy. In this case, we can write it as: \(W = K_f - K_i\) Where W is the work done, \(K_f\) is the final kinetic energy, and \(K_i\) is the initial kinetic energy. We know that the initial velocity of the gold nucleus is \(0.4243c\), the final speed is \(v_f\), the work done is \(140.779\, \rm{GeV}\), and the rest mass \(m_0\) is \(183.473\, \rm{GeV}/c^2\).

Calculate the initial kinetic energy

Using the relativistic kinetic energy formula, we have: \(K_{i} = (\gamma_{i} - 1)m_0c^2\) Where \(\gamma_{i}\) is the initial Lorentz factor, given by: \(\gamma_{i} = \frac{1}{\sqrt{1 - (0.4243c)^2/c^2}}\) Calculating the Lorentz factor: \(\gamma_{i} \approx 1.106\) Now, we can find the initial kinetic energy: \(K_{i} \approx (1.106 - 1)(183.473\, \rm{GeV}/c^2)c^2 \approx 20.243\, \rm{GeV}\)

Calculate the final kinetic energy

Using the work-energy theorem, we can find the final kinetic energy: \(K_{f} = K_{i} + W \approx 20.243\, \rm{GeV} + 140.779\, \rm{GeV} \approx 161.022\, \rm{GeV}\)

Relate final kinetic energy to final velocity

Now, we want to find the final velocity \(v_f\). We know the final kinetic energy and can relate this to the final velocity using the relativistic kinetic energy formula: \(K_{f} = (\gamma_{f} - 1)m_0c^2\) Where \(\gamma_{f}\) is the final Lorentz factor, given by: \(\gamma_{f} = \frac{1}{\sqrt{1 - v_f^2/c^2}}\) Solving for \(\gamma_{f}\): \(\gamma_{f} = \frac{K_{f}}{m_0c^2} + 1 \approx \frac{161.022\, \rm{GeV}}{183.473\, \rm{GeV}/c^2} + 1 \approx 1.877\)

Calculate the final velocity

Now, we can solve for the final velocity \(v_f\) using the Lorentz factor: \(v_f = c\sqrt{1 - \frac{1}{\gamma_{f}^2}}\) Plugging in the values, we get: \(v_f \approx c\sqrt{1 - \frac{1}{(1.877)^2}} \approx 0.8833c\) Therefore, the final speed of the gold nucleus as a fraction of \(c\) is approximately \(0.8833c\).

Key concepts

Work-Energy Theorem

The work-energy theorem is a fundamental principle in physics that relates the work done on an object to the change in its kinetic energy. To understand kinetic energy, imagine any moving object; it possesses energy due to its motion. If you apply a force to this object and move it, you're doing work on it, and this changes its energy.

Mathematically, it's expressed as W = K_f - K_i, where W is the work done, K_f is the final kinetic energy, and K_i is the initial kinetic energy. When it comes to relativistic speeds—those close to the speed of light—this theorem should include the effects of special relativity.

Lorentz Factor

The Lorentz factor, often denoted as γ, is crucial for understanding how objects behave as they approach the speed of light. It pops up in many equations of special relativity, including those that describe time dilation, length contraction, and relativistic momentum.

The Lorentz factor equation is: γ = 1 / √(1 - v²/c²), where v is the velocity of the object and c is the speed of light. As the speed of an object increases toward the speed of light, the Lorentz factor increases significantly, leading to noticeable relativistic effects. For example, as the Lorentz factor increases, the relativistic kinetic energy also increases, which must be accounted for in calculations like the one in the exercise.

Rest Mass

Rest mass, sometimes referred to as invariant mass, is the mass that an object possesses when it is at rest. It's a fundamental property of any object with mass, unaffected by the object's speed or position in space. In the realm of relativistic physics, rest mass plays a key role as it is the mass used in equations related to energy and momentum.

In our exercise example, the gold nucleus has a rest mass of 183.473 GeV/c². It's crucial to note this mass doesn't change regardless of the speed of the gold nucleus. However, as the velocity gets closer to the speed of light, the effective or relativistic mass would increase, which is a result of the kinetic energy contributing to the total energy of the particle.

Speed of Light

The speed of light, denoted as c, is a constant at approximately 299,792,458 meters per second. It's not just the speed at which light travels; it's also the maximum speed at which all energy, matter, and information in the universe can travel. According to Einstein's theory of special relativity, nothing can travel faster than the speed of light in a vacuum.

In our calculations, c is used as a scalar for velocities to express them as a fraction of the speed of light, such as the initial velocity of the gold nucleus being 0.4243c. The concept of the speed of light ties deeply into the work-energy theorem and Lorentz factor, especially when dealing with relativistic speeds that are significant fractions of c.