Problem 81 - Fundamentals of Physics

Question

A particle with mass \(m\) has speed \(c / 2\) relative to inertial frame S. The particle collides with an identical particle at rest relative to frame \(S .\) Relative to \(S,\) what is the speed of a frame \(S^{\prime}\) in which the total momentum of these particles is zero? This frame is called the center of momentum frame.

Step-by-Step Solution

Answer

The speed of frame \( S' \) is \( \frac{c}{4} \).

Step 1: Identify Initial Conditions

We have two particles with identical mass \( m \). The first particle, \( P_1 \), has an initial velocity \( v_1 = \frac{c}{2} \) and the second particle, \( P_2 \), is at rest, so \( v_2 = 0 \), relative to frame \( S \).

Step 2: Understand Total Momentum in Frame S

In frame \( S \), the total momentum of the system is given by the sum of the momenta of both particles. So, the total momentum is \( p_{total} = m v_1 + m v_2 = m \frac{c}{2} + 0 = \frac{mc}{2} \).

Step 3: Define Conditions for the Center of Momentum Frame

In the center of momentum frame \( S' \), the total momentum should be zero. This means that the velocity of the center of momentum frame, \( V_{cm} \), relative to frame \( S \), should offset the momentum of the two particles.

Step 4: Calculate the Velocity of the Center of Momentum Frame

The velocity \( V_{cm} \) should satisfy \( m(v_1 - V_{cm}) + m(v_2 - V_{cm}) = 0 \). Simplifying, \( m \left( \frac{c}{2} - V_{cm} \right) + m(0 - V_{cm}) = 0 \). Factoring out and solving gives\( m \left( \frac{c}{2} - 2V_{cm} \right) = 0 \). Thus, \( V_{cm} = \frac{c}{4} \).

Key Concepts

Momentum Conservation

Momentum conservation is a fundamental principle in physics. It tells us that the total momentum of a closed system remains constant if no external forces act on it. Momentum is the product of an object's mass and velocity. In mathematical terms, it is expressed as: \[ p = mv \] where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity.
In the exercise, two identical particles are involved. At first glance, it might seem unimportant, but because they have the same mass, calculations become simpler.
Initially, the total momentum of the system in frame \( S \) is calculated by adding the momentum of each particle: \[ p_{total} = mv_1 + mv_2 \]
When dealing with isolated systems, physicists often look for a frame where the total momentum becomes zero. This special frame simplifies many calculations and helps reveal intrinsic properties of the system.
For any collision or interaction, examining the momentum conservation helps us understand how objects interact, how they change speed, and what happens during and after the interaction.

Inertial Frame

An inertial frame is a reference frame where Newton's laws of motion hold true without the need for corrections due to non-inertia, like acceleration. In simpler terms, it is a non-accelerating frame where objects tend to stay at rest or in constant motion unless acted on by a force.
In our problem, two frames are particularly important: frame \( S \) and frame \( S' \). Frame \( S \) is stationary with respect to one particle and is an inertial frame. This allows us to easily apply Newton's laws to determine initial conditions like velocity and momentum.
Frame \( S' \), the center of momentum frame, is also inertial, but its speed relative to \( S \) is chosen such that the total momentum in \( S' \) is zero. This does not mean the particles stop moving in frame \( S' \), but rather their velocities balance out, making calculations more straightforward.
Choosing the right inertial frame is crucial to simplifying complex physics problems, especially involving momentum and collisions.

Particle Collision

A particle collision is an interaction where two or more particles come close enough to exert forces on each other, often leading to a transfer of energy and momentum. Collisions can be elastic, where kinetic energy is conserved, or inelastic, where kinetic energy is not conserved.
In this exercise, the collision is analyzed through momentum considerations. We are not given specifics about elastic or inelastic nature, but rather focus on how momentum is redistributed as a result of the collision.
To understand the unique perspective of the collision, we analyze it in the center of momentum frame \( S' \). Here, the total momentum before and after the collision is zero. With this condition, calculations and predictions about particle paths and speeds post-collision become more evidently clear and manageable.
Collisions are a key topic in physics, allowing deeper insights into particle interactions, energy transformation, and underlying forces affecting interacting bodies.