Question

A proton with mass \(1.673 \cdot 10^{-27} \mathrm{~kg}\) is moving with a speed of \(1.823 \cdot 10^{6} \mathrm{~m} / \mathrm{s}\) toward an alpha particle with mass \(6.645 \cdot 10^{-27} \mathrm{~kg}\), which is at rest. What is the speed of the center of mass of the proton-alpha particle system?

Short answer

Answer: The speed of the center of mass is approximately 3.665 * 10^5 m/s.

Step-by-step solution

Identify given values

We are given the mass and velocity of the proton (m1 and v1) and the mass of the alpha particle (m2). The alpha particle is at rest, so its velocity, v2, is 0. The given values are: m1 = 1.673 * 10^{-27} kg (mass of proton) v1 = 1.823 * 10^6 m/s (velocity of proton) m2 = 6.645 * 10^{-27} kg (mass of alpha particle) v2 = 0 m/s (velocity of alpha particle, at rest)

Apply the Center of Mass Velocity formula

Use the Center of Mass Velocity (CMV) formula: CMV = (m1 * v1 + m2 * v2) / (m1 + m2) Plug in the given values into the formula: CMV = ( (1.673 * 10^{-27} kg) * (1.823 * 10^6 m/s) + (6.645 * 10^{-27} kg) * (0 m/s) ) / ( (1.673 * 10^{-27} kg) + (6.645 * 10^{-27} kg) )

Solve for CMV

Now perform the calculations to find the speed of the center of mass: CMV = ( (1.673 * 10^{-27} kg) * (1.823 * 10^6 m/s) ) / ( (1.673 * 10^{-27} kg) + (6.645 * 10^{-27} kg) ) CMV = (3.053019 * 10^{-21} kg * m/s) / (8.318 * 10^{-27} kg) CMV ≈ 3.665 * 10^5 m/s The speed of the center of mass of the proton-alpha particle system is approximately 3.665 * 10^5 m/s.

Key concepts

Classical Mechanics

In the realm of physics, classical mechanics is a foundational theory that describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical bodies, such as spacecraft, planets, stars, and galaxies. It provides us with the tools necessary to predict how objects will move under the influence of various forces.

Understanding classical mechanics involves grasping the principles of kinematics (describing motion) and dynamics (the causes of motion), and applying laws formulated by Sir Isaac Newton. Newton's three laws provide the framework for analyzing an object’s motion, where the first law defines inertia, the second relates force to the acceleration of an object, and the third covers the principle of action and reaction.

One of the crucial applications of classical mechanics in problem-solving is determining the movements of two interacting bodies, which is often seen in collision exercises and conservation of momentum.

Momentum Conservation

The principle of momentum conservation is a cornerstone of physics that emerges from Newton's laws. This principle asserts that without external forces, the total momentum of a closed system remains constant through time. Momentum, a vector quantity, is the product of an object's mass and velocity (p = mv), and it's a measure of the 'quantity of motion' an object has.

In the context of our exercise, when we consider the proton and the alpha particle together as a system, the principle of momentum conservation allows us to say accurately that the system's total momentum before any interaction (such as collision) is equal to the total momentum afterward. This is immensely useful in analyzing collisions, explosions, and other interactions where no external forces are acting on the system. The momentum conservation principle simplifies problem-solving in classical mechanics and elucidates the behavior of isolated systems.

Particle Collision

Understanding particle collisions in physics is critical for dissecting interactions at both macroscopic and subatomic levels. When particles collide, their paths and velocities change in accordance with the laws of classical mechanics, particularly conservation of momentum.

There are two primary types of collisions: elastic collisions, where both momentum and kinetic energy are conserved, and inelastic collisions, where momentum is conserved, but kinetic energy is not. The solution provided deals with the speed of the center of mass, which is a concept often used in the analysis of inelastic collisions.

For two particles, the center of mass velocity is a weighted average of their velocities. Even when two particles collide and stick together, the center of mass moves at a constant velocity both before and after the collision, illustrating the conservation of momentum. This understanding of how particles interact during collisions is essential for advancing in physics problem solving.

Physics Problem Solving

Effective physics problem solving hinges on a systematic approach: identify the known information, determine the unknowns, select the relevant physical principles, and apply appropriate equations to solve for the unknowns.

In our exercise, we highlight these steps clearly. First, we identified the given mass and velocity values, understood the system interaction, and recognized that we needed to calculate the center of mass velocity (CMV). We then selected the appropriate formula that relies on the concept of momentum conservation. Lastly, we substituted our known values into the equation and simplified to solve for the unknown CMV.

This structured methodology not only helps in solving textbook exercises but also fosters a deeper understanding of the concepts involved, improving analytical skills necessary for encountering novel and complex problems in physics.