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Chapter 26: Problem 58

Show that Eq. \((26-11)\) reduces to the nonrelativistic relationship between momentum and kinetic energy, \(K \approx p^{2} /(2 m),\) for \(K<E_{0}\).

Short Answer

Expert verified
Question: Show that the given equation, Eq. (26-11), reduces to the nonrelativistic relationship between momentum and kinetic energy, \(K \approx p^2 / (2m)\) for \(K < E_{0}\). Answer: By substituting the total energy with the kinetic energy and rest energy, expanding and simplifying the equation, and using the nonrelativistic assumption, we derived the nonrelativistic relationship between momentum and kinetic energy, \(K \approx p^2 / (2m)\), valid for \(K < E_{0}\).

Step by step solution

01

Write down the given equation and definitions

The given equation is Eq. \((26-11)\), which is the relativistic relationship between the momentum \(p\), the rest energy \(E_{0}\), the kinetic energy \(K\), and the total energy \(E\): \[ E^2 = p^2c^2 + E_0^2. \] Here, \(E_0 = mc^2\) is the rest energy, and \(E = E_0 + K\) is the total energy, where \(m\) is the mass of the object and \(c\) is the speed of light.
02

Substitute the total energy with the kinetic energy

Now, let's substitute the total energy \(E\) with the kinetic energy \(K\) and the rest energy \(E_0\): \[ (E_0 + K)^2 = p^2c^2 + E_0^2. \]
03

Expand the left-hand side

Expand the square on the left-hand side of the equation: \[ E_0^2 + 2E_0K + K^2 = p^2c^2 + E_0^2. \]
04

Simplify the equation

Notice that the \(E_0^2\) terms cancel out on both sides of the equation. Thus, we are left with: \[ 2E_0K + K^2 = p^2c^2. \]
05

Use the nonrelativistic assumption

Under the nonrelativistic assumption, the kinetic energy \(K\) is much smaller than the rest energy \(E_{0}\) (\(K << E_0\)). Therefore, the \(K^2\) term is much smaller compared to the \(2E_0K\) term, and we can neglect \(K^2\): \[ 2E_0K \approx p^2c^2. \]
06

Divide by \(2E_0\) and use \(E_0 = mc^2\)

Divide both sides of the equation by \(2E_0\), and use the definition of rest energy \(E_0 = mc^2\): \[ K \approx \frac{p^2c^2}{2mc^2}. \]
07

Simplify and get the nonrelativistic relationship

Finally, cancel out the \(c^2\) terms: \[ K \approx \frac{p^2}{2m}. \] This is the nonrelativistic relationship between momentum and kinetic energy we needed to show, and it is valid for \(K < E_{0}\).

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