In some proton accelerators, proton beams are directed...

In some proton accelerators, proton beams are directed toward each other to produce head-on collisions. Suppose that in such an accelerator, protons move with a speed relative to the lab reference frame of $0.9972 c$ a) Calculate the speed of approach of one proton with respect to another one with which it is about to collide head on. Express your answer as a multiple of $c$, using six significant figures. b) What is the kinetic energy of each proton (in units of MeV) in the laboratory reference frame? c) What is the kinetic energy of one of the colliding protons (in units of $\mathrm{MeV}$ ) in the rest frame of the other proton?

In some proton accelerators, proton beams are directed toward each other to produce head-on collisions. Suppose that in such an accelerator, protons move with a speed relative to the lab reference frame of $0.9972 c$
a) Calculate the speed of approach of one proton with respect to another one with which it is about to collide head on. Express your answer as a multiple of $c$, using six significant figures.
b) What is the kinetic energy of each proton (in units of MeV) in the laboratory reference frame?
c) What is the kinetic energy of one of the colliding protons (in units of $\mathrm{MeV}$ ) in the rest frame of the other proton?

Key Concepts

Relativistic Velocity Addition

Relativistic velocity addition is a concept in special relativity that deals with combining velocities when objects move at speeds close to the speed of light. Unlike in Newtonian mechanics where velocities simply add, this principle uses a formula that ensures the resultant velocity never exceeds the speed of light. This is achieved by taking into account the effects of time dilation and length contraction, thereby preserving the invariant speed c in all reference frames.

Lorentz Factor

The Lorentz factor, commonly denoted as gamma (?), is a measure of the effects of time dilation, length contraction, and relativistic mass increase experienced by objects in motion relative to an observer. It is defined by the equation ? = 1/?(1 - v²/c²), where v is the velocity of the object and c is the speed of light. This factor is fundamental in translating classical expressions of time, distance, and energy into their relativistic equivalents.

Relativistic Kinetic Energy

Relativistic kinetic energy extends the classical concept of kinetic energy to situations involving high velocities near the speed of light. It is given by the formula K = (? - 1)mc², which accounts for the significant amounts of energy required to accelerate objects as they approach relativistic speeds. This concept is essential in the context of high-energy particle collisions, where the kinetic energy contributions are substantial due to relativistic effects.

Reference Frames in Special Relativity

Reference frames are coordinate systems in which observations and measurements are made, and they play a crucial role in special relativity. The laws of physics, including those governing energy and momentum, must hold true in all inertial frames. Understanding how quantities transform between these frames using the Lorentz transformations is essential for accurately describing phenomena such as particle collisions and relativistic energy dynamics.

Transcript

00:01 In this question, we have a proton beam which is directed towards another to produce a head -on collision.

00:06 We're told in such an accelerator, protons move with a speed relative to the lab reference frame of 0 .9972c.

00:16 We want to calculate the speed of approach of one proton with respect to another when they're about to collide head -on.

00:22 We want to find the kinetic energy of each of the protons in mega -electron volts in the laboratory reference frame and then in the rest frame of the other proton.

00:30 So what we know is the speed of the proton with respect to another is given by u prime plus v over 1 plus u prime v over c squared.

00:40 We know the relativistic kinetic energy is given by gamma minus 1 mc squared, where gamma is the lorentz factor as shown here.

00:48 So what we need to do is calculate the speed of one proton with respect to another.

00:55 So doing this, we can see that ux and we're given that the speed of the speed.

01:03 Of each proton with respect to the laboratory frame is 0 .997 .2c.

01:08 So we can say that this is equal to 0 .9972c plus 0 .9972c, as both of the protons have the same speed, over 1 plus 0 .9972c, and we can square this because it's multiplied by itself here over c squared, which gives us an answer of 0 .9999 -9 -9 -6c, which is the speed of the protons with respect to the laboratory frame.

02:09 The next thing we want to find out is the kinetic energy off each proton beam in the laboratory frame.

02:15 We know the kinetic energy is given by this expression here.

02:19 So we can substitute in our numerical values...

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