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

Show that each of these statements implies that $v<c$ which means that $v$ can be considered a nonrelativistic speed: (a) $\gamma-1<<1 \text { [Eq. }(26-14)] ;$ (b) $K<<m c^{2}$ [Eq. $(26-15)] ;(\text { c }) p<m c[\text { Eq. }(26-16)] ;$ (d) $K \approx p^{2} /(2 m)$.

Key Concepts

Momentum in the Nonrelativistic Regime

The relativistic momentum p = mv? reduces to the classical expression p = mv when the speed v is small compared to c. The condition p < mc implies that the momentum is not sufficient to cause noticeable relativistic effects. Maintaining momentum below this threshold ensures that the linear relationship between momentum and velocity in classical mechanics remains a valid approximation.

Approximation of Kinetic Energy as p²/(2m)

In the nonrelativistic limit, the kinetic energy of a particle can be approximated by the classical expression K ? p²/(2m). This approximation emerges from expanding the exact relativistic energy-momentum relation in powers of (v/c) and retaining only the lowest-order term. It underscores how, under nonrelativistic conditions, the behavior of a particle adheres to Newtonian mechanics, linking kinetic energy directly to the square of the momentum divided by twice the mass.

Lorentz Factor in the Nonrelativistic Limit

The Lorentz factor, defined as ? = 1/?(1 - v²/c²), measures the time dilation and length contraction effects at high speeds. In the nonrelativistic limit, where the speed v is much smaller than the speed of light c, the factor ? is approximately 1 and its deviation (? - 1) is very small. This condition allows one to use a Taylor series expansion of ? and simplify relativistic expressions to their classical forms.

Kinetic Energy Relative to Rest Energy

In special relativity, the total energy of a particle includes both the rest mass energy (mc²) and the kinetic energy. When the kinetic energy K is much smaller than the rest energy (K << mc²), the system is in the nonrelativistic regime. This inequality justifies approximating the energy expressions by neglecting higher-order terms in v/c, making classical mechanics applicable.

Transcript

00:01 So for question 8, we have gamma minus 1, that is much smaller than 1, which means that gamma is much smaller than 2.

00:12 And we know gamma is equal to 1 over square rule 1 minus v square over c squared.

00:22 So this is really much smaller than 2, which means that square rule 1 minus v square over c squared.

00:33 Is much bigger than one half.

00:37 Then we'll have 1 minus b square over c square.

00:42 It's much bigger than 1 quarter.

00:48 And v.

00:49 Square over c square is much less than 3 over 4.

00:58 Then we will have b over c is much less than square root 3 over 2.

01:08 So v is much less than square root 3 over 2, c, which means that v is much less than c.

01:21 Okay.

01:22 For question b, it was given us k that is much less than mc square.

01:31 Okay.

01:37 And when okay, can be equal to gamma minus 1 times mc square.

01:46 And it's much smaller than mc squared.

01:49 As you can tell, mc square and mc square can be cancelled.

01:53 Therefore, we have gama minus 1, much less than 1.

01:58 Remember, this is the exact same one from question 8, which means that we can just say from question a, say from a, we will have k that is much smaller than mc squared will lead to v is much smaller than c.

02:22 Okay? and for question c, it was given, let me see, let me see, let me see, we see, we'll see, was, okay, p smaller than mc squared.

02:45 I'm sorry, not mc squared, it's mc.

02:49 And therefore, we know momentum p can be equal to gamma times mv, and less than mc.

03:00 As you can tell, m and m can be canceled out.

03:05 So gamma v is less than c.

03:12 Therefore we have gamma, it's smaller than c over v...

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