Question 26.3: The Relativistic Momentum of an Electron GOAL Contrast the c...
The Relativistic Momentum of an Electron
GOAL Contrast the classical and relativistic definitions of momentum.
PROBLEM An electron, which has a mass of 9.11 × 10^{-31} kg, moves with a speed of 0.750c. Find the classical (nonrelativistic) momentum and compare it with its relativistic counterpart p_{\mathrm{rel}}.
STRATEGY Substitute into the classical definition to get the classical momentum, then multiply by the gamma factor to obtain the relativistic version.
First, compute the classical (nonrelativistic) momentum with υ = 0.750c:
p=m v=(9.11\times10^{-31}\,\mathrm{kg})(0.750\times3.00\times10^{8}\,\mathrm{m/s})
=\;2.05\times10^{-22}\,\mathrm{kg}\cdot\mathrm{m/s}
Multiply this result by γ to obtain the relativistic momentum:
p_{\mathrm{rel}}={\frac{m v}{\sqrt{1~-~v^{2}/c^{2}}}}={\frac{2.05~\times~10^{-22}{\mathrm{~kg}}~\cdot~{\mathrm{m}}/s}{\sqrt{1~-~(0.750c/c)^{2}}}}
=\;3.10\times\;10^{-22}\,{\mathrm{kg}}\cdot{\mathrm{m/s}}
REMARKS The (correct) relativistic result is 50% greater than the classical result. In subsequent calculations no notational distinction will be made between classical and relativistic momentum. For problems involving relative speeds of 0.2c, the answer using the classical expression is about 2% below the correct answer.