Question

Particle 1, of mass \(m_{1}\), moving at \(0.8 c\) relative to the lab. collides head-on with particle 2 , of mass \(m_{2}\), mov ing at \(0.6 c\) relative to the lab. Afterward, there is a single stationary object. Find, in terms of \(m_{1},(a) m_{2}:\) (b) the mass of the final stationary ob ject: and (c) the change in kinetic energy in this collision.

Short answer

(a) The mass of particle 2, \(m_{2}\) is \(1.33 m_{1}\). (b) The mass of the final stationary object is \(2.67 m_{1}\). (c) The change in kinetic energy can be calculated using the formulas mentioned in the steps above.

Step-by-step solution

- Analyze the given information

Particle 1 is moving with a speed of \(0.8c\) and has a mass \(m_{1}\). Particle 2 is moving with a speed of \(0.6c\) and has a mass \(m_{2}\). After the collision, they form a single stationary object. We want to find the mass \(m_{2}\), the mass of the final object and the change in kinetic energy.

- Find the mass of particle 2, \(m_{2}\)

Conservation of momentum applies for both classical and relativistic physics. In this scenario it is written as \(m_{1}\cdot 0.8c + m_{2}\cdot 0.6c = (m_{1}+m_{2})\cdot 0\). Simplification yields \(m_{2} = 1.33 m_{1}\)

- Find the mass of the single stationary object

The mass of an object moving with velocity \(v\) in relativistic physics is given by \(m = m_{0}/\sqrt{1-(v^{2}/c^{2})}\), where \(m_{0}\) is the rest mass of the object. From step 2, we know \(m_{2} = 1.33 m_{1}\). Conservation of energy states that after collision, the total energy before collision equals the rest energy (mc^2) of the final stationary object. The rest energy of the combined mass is \((m_{1}+\frac{1.33m_{1}}{\sqrt{1-(0.6c)^{2}/c^{2}}})c^{2}\), which simplifies to \(2.67m_{1}c^{2}\). Thus, the mass of the single stationary object becomes \(2.67m_{1}\)

- Find the change in kinetic energy

The change in kinetic energy is the difference in total energy before and after collision. Before the collision, the total energy is the sum of the energies of particle 1 and 2 which is \(E_{i} = m_{1}c^{2}/\sqrt{1-(0.8c)^{2}/c^{2}} + m_{2}c^{2}/\sqrt{1-(0.6c)^{2}/c^{2}}\). After the collision, the total energy is the rest energy of the stationary mass calculated in step 3, namely \(E_{f} = (2.67m_{1})c^{2}\). The change in kinetic energy \(\Delta KE = E_{i} - E_{f}\). This can be calculated by substituting the respective values and simplifying.

Key concepts

Relativistic Momentum Conservation

The principle of momentum conservation takes on a new dimension when we're dealing with speeds close to the speed of light, as described by Einstein's theory of relativity. In relativistic physics, the momentum of an object increases disproportionately as its velocity approaches the speed of light, symbolized by the constant 'c'.

For particles moving at these high relativistic speeds, their momentum is not simply the product of mass and velocity as in Newtonian physics, but instead is given by the formula: \( p = \frac{mv}{\sqrt{1-(\frac{v}{c})^2}} \).

This equation reveals that as 'v' increases towards 'c', the denominator shrinks, resulting in a significant increase in momentum. Let's apply this in our scenario: Particle 1 with mass \(m_{1}\) moving at \(0.8c\), and Particle 2 with mass \(m_{2}\) at \(0.6c\), both have 'relativistic momentum'. When they collide and come to a halt, the conservation of momentum implies all of the initial momentum is contained within the final stationary object. Thus, we equate the sum of the particles' momenta before the collision to zero afterward to find \(m_{2}\), ensuring that total momentum — in this case, relativistic momentum — is conserved.

Relativistic Energy Conservation

Relativistic energy conservation is a direct consequence of Einstein's famous equation \(E=mc^2\), which introduces the concept that mass and energy are equivalent. In a closed system, the total energy, which includes kinetic energy, potential energy, and rest mass energy, remains constant.

In our collision problem, energy conservation tells us that the sum of kinetic and rest mass energies of the particles before impact must equal the rest mass energy of the single, stationary object post-collision. The kinetic energy that both particles had before the collision, due to their high speeds, contributes to the mass of the final object according to the equation \( E = \frac{mc^2}{\sqrt{1-(\frac{v}{c})^2}} \).

After equating the initial total energy with the final rest mass energy, \(E_{f}\), and solving for the final mass in terms of \(m_{1}\), we found it to be \(2.67m_{1}\). This result encompasses the enhanced kinetic energy that was initially present in the fast-moving particles.

Kinetic Energy Change in Collisions

In relativistic collisions, kinetic energy change is not always as intuitive as in classical mechanics. While the law of conservation of energy holds true, a significant portion of kinetic energy can be converted into mass, and vice versa.

The overall change in kinetic energy, \(\Delta KE\), is the difference in the system's total energy before and after the collision. Before the collision, there's a significant amount of kinetic energy due to the high velocities of our particles, given by their relativistic energies. The combined kinetic energies contribute, through \(E=mc^2\), to the rest mass of the new stationary object formed in the collision.

By calculating the total initial energy, \(E_i\), and the final rest mass energy, \(E_f\), we determine the change in kinetic energy with the formula: \(\Delta KE = E_{i} - E_{f}\). The solution involves substituting known values and simplifying to grasp the energic transformation during the relativistic collision.

Relativistic Mass Increase

A fascinating aspect of Einstein's theory is that as an object's velocity increases toward the speed of light, its mass effectively increases, a phenomenon known as 'relativistic mass increase'. The equation representing an object’s relativistic mass is \( m = \frac{m_{0}}{\sqrt{1-(v^2/c^2)}} \), where \(m_{0}\) is the mass of the object at rest, and 'v' is its velocity.

This means that as an object moves faster, the kinetic energy it contains causes its mass to grow, as seen from an observer at rest. The implications of this are illustrated in our exercise: after the high-speed collision, the final stationary object has a mass greater than the simple arithmetic sum of the individual masses of the particles before the collision.

The collision's conservation equations allowed us to determine the mass of the final object, which incorporated increased mass due to the high kinetic energies prior to the collision. To put it simply, some of the kinetic energy was 'locked' into the increased mass of the stationary object, a prime example of the mass-energy equivalence principle.