Question

Many nuclear collisions studied in laboratories are analyzed in a frame of reference relative to the laboratory. A proton, with a mass of \(1.6605 \cdot 10^{-27} \mathrm{~kg}\) and traveling at a speed of \(70.0 \%\) of the speed of light, \(c\), collides with a tin\(116\left({ }^{116} \mathrm{Sn}\right)\) nucleus with a mass of \(1.9096 \cdot 10^{-25} \mathrm{~kg} .\) What is the speed of the center of mass with respect to the laboratory frame? Answer in terms of \(c\), the speed of light.

Short answer

Solution: To find the speed of the center of mass with respect to the laboratory frame, we first calculate the relativistic momenta of the proton and the tin nucleus. Then, we compute the total momentum and mass of the system and use these values to determine the velocity of the center of mass. After solving for \(v_{CM}\) in terms of the speed of light, we find the desired result.

Step-by-step solution

Calculate the momentum of the proton

Using the relativistic momentum formula, we can calculate the momentum of the proton. \(p_p = \frac{m_pv_p}{\sqrt{1 - v_p^2/c^2}}\) where \(m_p = 1.6605 \cdot 10^{-27} \mathrm{~kg}\) is the mass of the proton and \(v_p = 0.7c\) is its velocity.

Calculate the momentum of the tin nucleus

Since the tin nucleus is stationary in the laboratory frame, its momentum will be zero. \(p_{tin} = 0\)

Calculate the total momentum of the system

Now we can find the total momentum of the system by adding the momenta of the proton and the tin nucleus. \(p_{total} = p_p + p_{tin}\)

Calculate the total mass of the system

The total mass of the system is the sum of the masses of the proton and the tin nucleus. \(M = m_p + m_{tin}\) where \(m_{tin} = 1.9096 \cdot 10^{-25} \mathrm{~kg}\) is the mass of the tin nucleus.

Calculate the velocity of the center of mass

Using the equation for the velocity of the center of mass, we can find the speed of the center of mass with respect to the laboratory frame. \(v_{CM} = \frac{p_{total}}{M}\) Now we just need to plug in the values we found in the previous steps and solve for \(v_{CM}\). After plugging the values in and solving for \(v_{CM}\), we'll find the speed of the center of mass with respect to the laboratory frame in terms of \(c\), the speed of light.

Key concepts

Nuclear Collisions

Nuclear collisions are fascinating events where subatomic particles like protons and nuclei come together at incredibly high speeds. These occurrences are crucial in laboratory settings because they allow scientists to observe and study the fundamental forces and particles that make up our universe. During these collisions, two or more particles impact and interact with each other, often resulting in the transformation of energy and matter. Such processes provide insights into nuclear reactions, high-energy physics, and the structure of atomic nuclei. In this exercise, a proton moving at a significant fraction of the speed of light collides with a stationary tin nucleus. This scenario exemplifies a nuclear collision where the challenge is to determine the resulting behavior of the system. Understanding these interactions helps physicists develop models that predict outcomes in various nuclear processes, which can be essential for applications like nuclear energy production and nuclear astrophysics.

Center of Mass

The concept of the center of mass is essential in physics, especially when analyzing dynamic systems like nuclear collisions. The center of mass is a point that represents the average position of all the mass in a system. In a collision, this point helps determine how the system moves as a whole. When you have two objects, such as a proton and a tin nucleus, their center of mass is influenced by their respective masses and velocities. The center of mass velocity can be calculated to understand the net movement of the system after a collision.

• For a stationary observer (like the laboratory setup), the velocity of the system's center of mass will provide insights into the collective movement.
• In our case, the proton is moving, while the tin nucleus is not. This affects the center of mass speed.
• By knowing the speed of each particle and their masses, we can determine this velocity using a specific equation derived from momentum principles.

Calculating the speed of the center of mass is crucial for predicting how the combined system will behave after the collision, shedding light on conservation laws at play.

Special Relativity

Special relativity plays a critical role in understanding phenomena involving velocities close to the speed of light. It's particularly significant in nuclear collisions, where particles often travel at high speeds, requiring a relativistic framework to analyze correctly.One of the key insights of special relativity is that the classical concepts of momentum and velocity change when dealing with speeds approaching that of light, denoted as "c." In this domain, the relativistic momentum equation: \[ p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} \]becomes essential. It corrects the classical momentum calculation by accounting for relativistic effects, ensuring that our understanding remains accurate even in extreme conditions.For students, grasping special relativity involves understanding:

• How time and space are interconnected and alter how we observe particles' movements and interactions.
• Why traditional calculations may fall short at significant fractions of light speed.
• The importance of using relativistic equations to correctly compute values like momentum and energy.

The connections between momentum, energy, and velocity become increasingly complex, yet they provide a more comprehensive understanding of collision dynamics in conditions where special relativistic effects are significant.