Question

Consider the collisions of two identical particles. each of mass \(m_{0}\). In experiment \(A\), a particle moving at \(0.9 c\) strikes a stationary particle. (a) What is the total kinetic energy before the collision? (b) In experiment \(B\), both particles are moving at a speed \(u\) (relative to the lab), directly roward one another. If the total kinetic energy bef ore the collision in experiment \(B\) is the same as that in experiment A, what is \(u\) ? c) In both experiments, the particles stick together. Find the mass of the resulting single particle in each experiment. In which is more of the initial kinetic energy converted to mass?

Short answer

Therefore, to find the total kinetic energy before collision in experiment \(A\), the speed in experiment \(B\) when the total kinetic energy is same as in \(A\), and the mass of the resulting particle in each experiment, the concept of relativistic kinetic energy and mass-energy equivalence principle need to be utilized and the resulting equations solved correctly.

Step-by-step solution

Calculate the total kinetic energy before the collision in experiment A

The kinetic energy of a relativistic particle is given by \(K = (\gamma - 1) m_0 c^2\), where \(\gamma\) is the Lorentz factor given by \(\gamma = 1/\sqrt{1-(v^2/c^2)}\). Therefore, the total kinetic energy in experiment \(A\) is \(K_A = (\gamma - 1) m_0 c^2 + 0\), since the second particle is stationary.

Determine the speed in experiment B

In experiment \(B\), both particles are moving towards each other with speed \(u\). Therefore, their total kinetic energy is \(2(\gamma_u - 1) m_0 c^2\) where \(\gamma_u = 1/\sqrt{1-(u^2/c^2)}\). Setting this equal to \(K_A\) and solving for \(u\) will give the required speed.

Calculate the mass of the resulting particle in each experiment

When the particles stick together, the mass of the resulting particle is given by the energy conservation principle, which states that the total energy before the collision is equal to the total energy after. The energy before the collision includes the kinetic energy and rest energy (\(m_0 c^2\)) of each particle. In experiment \(A\), the total energy before the collision is \(K_A + 2 m_0 c^2\). Therefore, the mass \(m_A\) of the resulting particle in experiment \(A\) is given by \(m_A c^2 = K_A + 2 m_0 c^2\), which can be solved for \(m_A\). Similar calculation can be done for experiment \(B\).

Determine which experiment has more kinetic energy converted to mass

This can be determined by comparing the masses \(m_A\) and \(m_B\) obtained from step 3. The experiment with more increase in mass implies more conversion of kinetic energy to mass.

Key concepts

Kinetic Energy in Special Relativity

Kinetic energy in classical mechanics is simple: it's \frac{1}{2}mv^2\frac{1}{2}mv2 , where m is mass and v is velocity. However, when objects move at speeds close to the speed of light (denoted as c), we must use special relativity to accurately describe their behavior. In special relativity, the kinetic energy (K) of an object moving at a significant fraction of c is described by the equation: K = (\text{\gamma} - 1)m_{0}c^2where \(m_{0}\) is the rest mass of the object, \(c\) is the speed of light in a vacuum, and \(\text{\gamma}\) is the Lorentz factor. This formula reveals that at high speeds, the kinetic energy grows much more rapidly than it would under the classical expression, due to the Lorentz factor's influence.In the given exercise, we compute the total kinetic energy before the collision in experiment A using this relativistic formula, to see how it differs from the classical approach and implies a much larger kinetic energy than what might be expected at such high velocities.

Lorentz Factor

The Lorentz factor is critical in understanding how time, space, and energy behave at high velocities. It's defined as:\(\gamma = \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}\)This factor becomes significant as an object's speed v gets close to c, causing time dilation and length contraction as predicted by Einstein's theory of special relativity. When \(v\) is much less than \(c\), \(\gamma\) is approximately 1, and the classical mechanics apply. However, as \(v\) approaches \(c\), \(\gamma\) increases dramatically, affecting the object's relativistic mass, time perception, and kinetic energy.In the exercise, we see that the Lorentz factor directly affects the total kinetic energy of particles moving at relativistic speeds and thus the dynamics of collisions at such high velocities.

Energy Conservation in Particle Physics

Conservation of energy is a fundamental principle in physics that holds true even when we examine events on the particle level. However, in the realm of high-energy physics, energy conservation must account not only for kinetic and potential energy, but also the equivalent energy of mass, as described by Einstein’s famous equation \(E=mc^2\).During relativistic collisions, such as those in our exercise, energy can convert into mass and vice versa. Thus, when two particles collide and combine, the total energy – which includes rest mass energy and kinetic energy – before the collision must equal the total energy – now entirely rest mass energy – of the resulting particle. This principle allows us to calculate the rest mass of the combined particle after such a collision occurs, offering deep insights into the processes governing particle interactions.

Mass-Energy Equivalence

Mass-energy equivalence is one of the most famous concepts in physics, encapsulated in Albert Einstein's equation \(E=mc^2\). This equation tells us that mass and energy are two forms of the same thing and can be converted into each other. In relativistic collisions, kinetic energy can be transformed into mass, often observed in particle accelerators where the kinetic energy of colliding particles creates new particles with mass.In the context of our exercise, mass-energy equivalence allows us to understand how kinetic energy can convert into the mass of a new, larger particle when two smaller particles collide and stick together. Comparing the masses of the resulting particles in different experiments shows us how much kinetic energy has been converted into mass, as seen in experiments A and B where the collision outcomes allow us to quantify this transformation through the results of the conservation of energy.