A particle decays in flight into two pions, each having a rest energy of 140.0 MeV. The pions tr

A particle decays in flight into two pions, each having a...

Key Concepts

Conservation of Relativistic Momentum

In relativistic physics, momentum is conserved in all isolated systems, even when particles travel at speeds comparable to that of light. This requires taking into account both the magnitude and direction of momentum, and using the relativistic definition p = ?mv, where ? is the Lorentz factor. Conservation of momentum is crucial in decay processes, where the momentum of the original particle is related to the vector sum of the momenta of its decay products.

Relativistic Energy-Momentum Relation

This relation, given by E² = (pc)² + (m?c²)², connects the energy, momentum, and rest mass of a particle. It is central to analyzing high-speed particles and decay events because it allows one to solve for unknown quantities like kinetic energy and invariant mass using the measurable energy and momentum of decay products.

Lorentz Factor

The Lorentz factor, defined as ? = 1/?(1 - v²/c²), plays a critical role in relativistic mechanics to account for time dilation and length contraction effects at high velocities. It is used to calculate both the relativistic momentum and kinetic energy, distinguishing relativistic calculations from their classical counterparts.

Invariant Mass

Invariant mass, sometimes referred to as rest mass, remains constant regardless of the reference frame and is calculated using the total energy and momentum of the system. In decay processes, the invariant mass of the original particle can be determined from the sum of the four-momenta of its decay products, making it a fundamental quantity in particle physics.

Transcript

00:01 Okay, so for question ahead, let's find out the momentum for the particle.

00:04 Well, we know due to conservation law for the momentum, we have momentum of the particle is equal to the combination of the momentum from the pions, okay, which is p -py, plus p -py.

00:24 And if we put square here, we'll have p -e -square is equal to p -py plus p -py square.

00:32 The reason why i put square here instead of just use p equals 2 p pi is because there are angles, there's angles between two pylons, okay? which is right angles and this will give us p square is equal to p pi plus p pi square which is p square is equal to you know what, let me do this p pi square plus p pi square and then plus two times p pi pi p pi, n times cosine angle which is 90 degrees.

01:23 And we know that cosine 90 degrees is 0.

01:26 Therefore, p squared is equal to p square pi plus p square pi.

01:37 It's because cosine 90 degree, which is 0, and 0 times everything is anything is 0.

01:42 Therefore, just give us such a question here.

01:44 Then we have p square is equal to 2 p square pi.

01:50 And then if we put square root here, we'll have p is just simply equal to square root two p pi.

02:05 And we know that p is equal to square root to p pi, and the momentum of the pion can be expressed as gamma, let's say m pi, v, okay? which can be extended to such equation, which is p is equal to square rule 2 times gamma, which is 1 over square rule, 1 minus v square, over c squared, n times m pi, times v, which is given as 0 .90c.

02:59 And then if we keep doing that, we'll have, let's plug in the v value, okay? for the gamma, the lorentz factor, we have square 2 times 1 over square root 1 minus 0 .90c square over c squared times m pi times 0 .90c.

03:27 And we know the rest energy for the pion, let's say e0 pi, is equal to mc square.

03:39 Okay, and therefore, if we move 1c to the left, i will have e0 pi divide by c, which is equal to mc.

03:54 And since we know that mc is equal to e0 pi divide by c, and e0 pi, which is a rest energy for a pion, it's about 140 mega electron volts.

04:07 And since it's per c, it means it's 140 megavolts over c.

04:17 And let's take a look at the equation here.

04:19 If we do some arrangement, we have p is equal to square root 2 times remember since it's 1 over square root 1 minus 0 .90 c squared over c squared there will have 1 over square root 0 .19 okay because 0 .90 square is 0 .81 and 1 minus 0 .81 is 0 .1 and c square will eventually cancel out so there's why there's no c square there and times 0 .90 times m -p -c.

04:57 I'm sorry, i forgot to put the pi under the m here.

05:05 So therefore, as you can tell, we can just simply plug in the weather here back in here.

05:15 And then we can determine the momentum for the particle, which is p, is just simply equal to square root 2 times 1 over square root 0 .19 times 0 .90, and n times 14, 140.

05:36 Mega electron volts per c and this will give us the momentum is equal to let me see 409 mega electron volts per cool i'm sorry per c per speed of light okay and let's take a look at question b which is who was asking us the magnitude of a kinetic energy well first let's take look at the total energy when our total energy is equal to two times the total energy of a pylon.

06:19 And this is according to the energy conservation law.

06:24 And then if we put square, we'll have e square.

06:35 It's just simply equal to four times e pi square.

06:41 Okay.

06:45 And we know that the total energy for pylon is equal to the rest energy of a pion plus kinetic energy...

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