Show that each of these statements implies that v ≪c which means that v can be considered a nonr

Show that each of these statements implies that v ≪c which...

Key Concepts

Lorentz Factor

The Lorentz factor, denoted by ?, quantifies the effects of time dilation, length contraction, and relativistic mass variation in special relativity. It is defined as ? = 1/?(1 ? v²/c²) and is central to the transformation between inertial frames. In the nonrelativistic limit (v ? c), a series expansion shows that ? is approximately 1 + ½(v²/c²), indicating that relativistic corrections are very small and classical mechanics suffices.

Kinetic Energy in Relativity

Relativistic kinetic energy is given by K = (? ? 1)mc², which captures both the rest energy and the additional energy due to motion. When v is much smaller than c, the approximation ? ? 1 + ½(v²/c²) makes the kinetic energy approach the classical form, K ? ½mv². This demonstrates that if the kinetic energy is much less than the rest energy (mc²), the system is effectively in the nonrelativistic regime.

Relativistic Momentum

In relativity, momentum is expressed as p = ?mv, incorporating the Lorentz factor to account for high-speed effects. For speeds much less than the speed of light, the factor ? is nearly unity, so p approximates mv. The condition p ? mc ensures that additional relativistic momentum contributions are negligible, reaffirming the validity of using classical momentum expressions when v ? c.

Nonrelativistic Limit Approximations

The nonrelativistic limit refers to the regime where the velocity v of a particle is much smaller than the speed of light c. In this limit, all relativistic quantities such as the Lorentz factor, kinetic energy, and momentum reduce to their classical counterparts through series expansions that neglect higher order terms in v²/c². This approximation underpins the consistency between classical mechanics and special relativity at low speeds.

Transcript

00:01 In this problem, we are looking at the different ways that we basically are saying that v is much less than c, so that the non -relativistic equations could be used, that we're in the non -relativistic realm.

00:15 So let's look at each of these expressions in c.

00:17 It actually does tell us that v is much less than c.

00:21 So first one is gamma minus one much less than one.

00:25 And so that gives us gamma minus 1, which is equal to 1 over a square root, 1 minus 1.

00:31 V squared or c squared minus one is much less than one.

00:42 And this tells me then that one over the square root of one minus v squared over c squared is much less than two.

00:53 You manipulate, don't let these greater than signs, these inequality signs, approximation signs, any of these things, bother you.

01:02 Treat them, treat it just like an equal, sign and just maintain the order.

01:08 So this tells me that that one over square root of 1 minus v squared or c squared is much less than two.

01:16 This is the same as gamma being much less than two.

01:21 I'm going to need that in part c.

01:23 So i'm just going to make a note of that here.

01:25 Don't worry about it for this part.

01:27 Okay.

01:29 So now let's multiply both sides by the square root and divide both sides by 2, and that's going to give us an n -square, one -quarter is much less 1 -5 -0 -1 -5 % over c -squared.

01:50 This implies v -c -squared over -c -squared is much less than 1 -1 -1 -quarter, which implies v -c -c -squared is much less than three quarters, which implies v is much less than square root of three over two times c.

02:16 Now, square root of three over two is 0 .866.

02:21 So that's already less than c.

02:23 So this quantity is less than c.

02:30 So that implies that v is much less than c.

02:36 If you're already less than something that's a fraction of c, then obviously you're much less than c itself.

02:44 So you have that.

02:48 So that is part a, that's less than c.

02:55 That we show that that, if you saw this criteria, if you saw this expression, then you know that you're in the non -relativistic round, v much less than c.

03:08 Part b, this expression, k much less, than c or this condition, if you like.

03:21 So let's put in our expression for relative relative relative energy, gamma minus 1, mc squared is much less than mc squared.

03:31 The mc squared, go away.

03:33 This implies gamma minus 1 is much less than 1.

03:38 Well, c, c, a above.

03:47 So that proves that one, that it does indicate that v is much less than c.

03:56 Part c.

03:58 P much less than mc so m v gamma is much less than mc so we get from this v over c is much less than one over gamma so this gives me that v is much less than c over gamma now let's talk about gamma and i'm going to talk a more than what i need.

04:32 From what i wrote in part a, gamma in these, when we're talking about the small, you know, v much less than c realm, the non -respected realm, is much less than two.

04:45 We saw that in part a.

04:47 That implies that one over a gamma is much greater than one -half...

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