183_notes:examples:momentumfast

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183_notes:examples:momentumfast [2014/07/10 14:20] – caballero183_notes:examples:momentumfast [2024/01/30 14:18] (current) – [Setup] hallstein
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 ====== Example: Calculating the momentum of a fast-moving object ====== ====== Example: Calculating the momentum of a fast-moving object ======
  
-An electron is observed to be moving with a velocity of $\langle -2.05\times10^7, 6.02\times10^7, 0\rangle\:\dfrac{m}{s}$. Determine the momentum of this electron.+An electron is observed to be moving with a velocity of ⟨−2.05×107,6.02×107,0⟩ms. Determine the momentum of this electron.
  
 ==== Setup ==== ==== Setup ====
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   * An electron is in motion   * An electron is in motion
-  * It has a velocity of $\langle -2.05\times10^7, 6.02\times10^7, 0\rangle\:\dfrac{m}{s}$. +  * It has a velocity, $\vec{v}_e=\langle -2.05\times10^7, 6.02\times10^7, 0\rangle\dfrac{m}{s}$. 
-  * This velocity is near the speed of light (c=3.00×108ms).+  * The speed of the electron is near the speed of light (c=3.00×108ms).
  
 === Lacking === === Lacking ===
  
-  * The mass of the electron is not given, but can be [[http://lmgtfy.com/?q=electron+mass|found online]] (me=9.11×10−31kg).+  * The mass of the electron is not given, but can be [[https://en.wikipedia.org/wiki/Electron_mass|found online]] (me=9.11×10−31kg).
  
 === Approximations & Assumptions === === Approximations & Assumptions ===
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 First, we compute the speed of the electron. First, we compute the speed of the electron.
  
-|→v|=√v2x+v2y+v2z=√(−2.05×107ms)2+(6.02×107ms)2+(0)2=6.36×107ms+$$|\vec{v}_e| = \sqrt{v_x^2+v_y^2+v_z^2} = \sqrt{(-2.05\times10^7 \dfrac{m}{s})^2+(6.02\times10^7 \dfrac{m}{s})^2+(0)^2} = 6.36 \times 10^7 \dfrac{m}{s}$$
  
 Next, we compute the gamma factor. Next, we compute the gamma factor.
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 Finally, we compute the momentum vector. Finally, we compute the momentum vector.
  
-$$\vec{p} = \gamma m \vec{v} = (1.02) (9.11\times10^{-31} kg) \langle -2.05\times10^7, 6.02\times10^7, 0\rangle\:\dfrac{m}{s} = \langle \rangle\:\dfrac{kg\:m}{s}$$+$$\vec{p}_e = \gamma m_e \vec{v}_e = (1.02) (9.11\times10^{-31} kg) \langle -2.05\times10^7, 6.02\times10^7, 0\rangle\dfrac{m}{s} = \langle -1.91 \times 10^{-23}, 5.61\times10^{-23},0\rangle\dfrac{kg\:m}{s}$$
  
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  • Last modified: 2014/07/10 14:20
  • by caballero