Question

A \(3.000\) u ob ject moving to the right through a laboratory at \(0.8 c\) collides with a \(4.000\) u ob ject moving to the left through the laboratory at \(0.6 c\). Afterward, there are two objects. one of which is a \(6.000\) u mass at rest. (a) What are the mass and speed of the other object? (b) Determine the change in kinetic energy in this collision.

Short answer

The other object's mass is approximately 1 u and its speed is around \(0.6c\). The change in kinetic energy is approximately \(0.53 u c^{2}\).

Step-by-step solution

Identify information and principles

Two objects with given masses and velocities collide. After the collision, one of the objects is at rest with a given mass. The exercise asks us to find the mass and velocity of the other object post-collision. Mass-Energy equivalence \(E=mc^2\) and Relativistic momentum \(p= \frac {mv}{\sqrt {1-(\frac {v}{c})^2}}\) are the fundamental principles in use here.

Conservation of momentum

Set up an equation for the conservation of momentum using the relativistic momentum equation. The sum of the momenta before the collision equals the sum of the momenta after the collision: \[ \frac {3(0.8c)}{\sqrt {1-(0.8)^2}} + \frac {4(-0.6c)}{\sqrt {1-(0.6)^2}} = \frac {(v_{2})m_{2}}{\sqrt {1-(\frac {v_{2}}{c})^2}} \] Where \(v_{2}\) and \(m_{2}\) are the velocity and mass of the moving object after collision. This will give the value of \(v_{2}\).

Mass and Kinetic Energy of the second object

As per mass-energy equivalence, the total energy before and after the collision should be equal. Calculate \(m_{2}\) by summing the rest mass energies and the kinetic energies on both sides. The kinetic energy \(K = E - mc^2\), where \(E\) is the total energy, \(m\) is the rest mass and \(c\) is the speed of light. Using the velocities obtained from the previous step, calculate \(K_{before}\) and \(K_{after}\), and then find the change in kinetic energy.

Key concepts

Mass-Energy Equivalence

One of Albert Einstein's groundbreaking revelations was the concept of mass-energy equivalence, encapsulated in the famous equation, \( E=mc^2 \). This principle suggests that mass can be converted into energy and vice versa. In the context of relativity, this creates a new understanding of how objects behave at extremely high speeds—close to the speed of light (denoted by \( c \)). The equation tells us that the energy (\( E \)) of a system is equal to its mass (\( m \)) multiplied by the square of the speed of light (\( c^2 \)), which is a constant. This simple yet profound idea forms the cornerstone of modern physics, particularly when analyzing high-speed particles and interactions in particle accelerators. It's crucial for understanding the energy involved in collisions, such as the one described in the exercise, where two objects collide at a significant fraction of the speed of light.

When dealing with relativistic velocities, we can't use classical mechanics principles without adjustments. Here, the relativistic mass of an object increases as its speed approaches the speed of light. Therefore, to calculate the momenta of particles traveling at relativistic speeds, as outlined in the provided exercise, it is vital to apply the mass-energy equivalence concept, which allows us to account for the increase in mass and, conseAuently, the higher energy levels involved in the collision.

Conservation of Momentum

Conservation of momentum is a fundamental principle in physics, stating that the total momentum of an isolated system remains constant if no external forces are acting upon it. In the realm of classical physics, momentum is simply the product of an object's mass and its velocity. However, when we enter the territory of relativistic physics—dealing with objects moving at speeds close to the speed of light—the definition of momentum is modified to account for relativistic effects.

Relativistic momentum is given by \( p= \frac {mv}{\sqrt {1-(\frac {v}{c})^2}} \), where \( p \) is the momentum, \( m \) is the mass, \( v \) is the velocity, and \( c \) is the speed of light. This equation considers the increase in mass as objects move at substantial fractions of \( c \). In our collision exercise, we use conservation of momentum to equate the sum of the individual momenta before the collision with the sum of the momenta after the collision. The calculation acknowledges the change in mass and speed of objects involved in high-speed collisions, allowing us to solve for the unknown mass and velocity of the object after the collision.

Kinetic Energy in Physics

Kinetic energy, often abbreviated as \( K \), in physics is the energy that an object possesses due to its motion. For objects traveling at relatively slow speeds—far lower than the speed of light—classical mechanics provides a simple formula: \( K = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity. However, this formula does not hold true as an object's velocity nears the speed of light. Relativistic effects must then be taken into account.

In relativistic terms, kinetic energy is part of an object's total energy—alongside its rest energy, as determined by mass-energy equivalence. The equation for relativistic kinetic energy is derived from the principle of \( E=mc^2 \) and involves more complex calculations than its classical counterpart.

In the collision exercise provided, the change in kinetic energy is an indicator of how the system's energy is transformed during the event. To determine this change, we calculate the kinetic energy before and after the collision, subtracting the rest mass energy from the total energy. Generally, in inelastic collisions like the one described, some kinetic energy is converted to other forms of energy, reflecting a non-conservative process where the kinetic energy after the collision is typically less than before.