A particle of mass m, initially moving with speed v,...

A particle of mass $m$, initially moving with speed $v$, collides head on elastically with an identical particle initially at rest. (a) Do you expect the total mass of the two particles after the collision to be (1) greater than $2 m$ (2) equal to $2 m,$ or (3) less than $2 m$ ? Why? (b) What are the total energy and momentum of the two particles after the collision, in terms of $m, v,$ and $c ?$

A particle of mass $m$, initially moving with speed $v$, collides head on elastically with an identical particle initially at rest. (a) Do you expect the total mass of the two particles after the collision to be (1) greater than $2 m$
(2) equal to $2 m,$ or (3) less than $2 m$ ? Why? (b) What are the total energy and momentum of the two particles after the collision, in terms of $m, v,$ and $c ?$

Key Concepts

Rest Mass Invariance

In an elastic collision, the intrinsic rest mass of each particle does not change. Although the particles exchange kinetic energy and momentum during the collision, no mass is converted to energy or created anew, so the total rest mass of the system stays the same. This concept is important in understanding that the combined rest mass remains constant at 2m for the two particles.

Conservation of Energy and Momentum

In any collision, especially an elastic collision, the fundamental principles of conservation of energy and momentum apply. This means that the total energy (which includes both rest energy and kinetic energy) and the total momentum of the system remain constant before and after the collision. These principles are crucial for analyzing the outcome of collisions both in classical mechanics and in special relativity.

Relativistic Energy and Momentum

When speeds are comparable to the speed of light, relativistic formulas must be used to account for the effects predicted by special relativity. The energy of a particle is given by E = ?mc² and its momentum by p = ?mv, where ? (the Lorentz factor) is defined as 1/?(1 ? v²/c²). Using these relationships allows one to correctly determine the total energy and the momentum of the particles after the collision when expressed in terms of m, v, and c.

Transcript

00:02 Hello, and in this question here, we've got a mass initially moving at a velocity v, which collides with another object, another particle which has the same mass of it, and then these two move off together with some new velocity v -subscript f for final.

00:18 And we want to ask, is the total mass of the new system, of the two particles together after the collision, greater than equal to or less than 2m? okay, well, this mass here, okay, there's some kinetic energy here.

00:37 So the kinetic energy that the system has, okay, we can relate this energy that due to kinetic motion, we can relate this to a mass by using e equals mc squared.

00:51 So if we have a positive kinetic energy from the motion, well, kinetic energy is always going to be positive, but if we have a positive energy from the motion, this would mean we'd have an increase in the mass.

01:04 So we would expect that the total mass, so mass total to be greater than 2m.

01:11 Sorry, then i wrote that down wrong, 2m0.

01:17 And now in the second part of the question, we actually want to find out what the momentum and the energy is in terms of the initial variables we've been given.

01:27 So to do this, we're going to apply conservation of energy and conservation of momentum.

01:33 And conservation of momentum, so the momentum is equal to the initial momentum of the system, which is gamma times m0 times v.

01:43 So this is the momentum of, let's scroll up, this is the momentum of this particle here.

01:50 Then we add zero for the momentum of this particle here, which is at rest.

01:55 And this equals the momentum after, which is equal to gamma times 2m0 times the new final velocity.

02:03 And we note that this gamma here is not the same as this gamma as this gamma is a function of v, and this gamma is a function of v subscript f.

02:12 Okay? so the final momentum is equal to m0 times v divided by the square root of 1 minus v squared over c squared.

02:27 Okay.

02:28 Now, the energy, okay? well, the energy is just equal to gamma times the rest mass energy.

02:35 Energy times c squared.

02:38 Now gamma is a function of v -subscript f, as this is the final velocity.

02:44 So it's equal to 2m0 times c squared, divided by the square root of one divided by, sorry, of one minus v -subscript f to be squared, divided by c squared...

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