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183_notes:examples:momentumfast [2014/07/10 13:50] – caballero | 183_notes:examples:momentumfast [2014/07/10 14:20] – [Solution] caballero | ||
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+ | ~~NOTOC~~ | ||
====== 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 2.05e7, 6.02e7, 0\rangle\: | + | An electron is observed to be moving with a velocity of $\langle |
+ | ==== Setup ==== | ||
+ | |||
+ | You need to compute the momentum of this electron using the information provided and any information that you can collect or assume. | ||
+ | |||
+ | === Facts ==== | ||
+ | |||
+ | * An electron is in motion | ||
+ | * It has a velocity of $\langle -2.05\times10^7, | ||
+ | * This velocity is near the speed of light ($c = 3.00\times10^8 \dfrac{m}{s}$). | ||
+ | |||
+ | === Lacking === | ||
+ | |||
+ | * The mass of the electron is not given, but can be [[http:// | ||
+ | |||
+ | === Approximations & Assumptions === | ||
+ | |||
+ | * The electron does not experience any interactions, | ||
+ | |||
+ | === Representations === | ||
+ | |||
+ | * The momentum of the electron is given by $\vec{p} = \gamma m \vec{v}$ where $\gamma = \dfrac{1}{\sqrt{1-\left(\dfrac{|\vec{v}|}{c}\right)^2}}$. | ||
+ | |||
+ | ==== Solution ==== | ||
+ | |||
+ | First, we compute the speed of the electron. | ||
+ | |||
+ | $$|\vec{v}| = \sqrt{v_x^2+v_y^2+v_z^2} = \sqrt{(-2.05\times10^7 \dfrac{m}{s})^2+(6.02\times10^7 \dfrac{m}{s})^2+(0)^2} = 6.36 \times 10^7 \dfrac{m}{s}$$ | ||
+ | |||
+ | Next, we compute the gamma factor. | ||
+ | |||
+ | $$\gamma = \dfrac{1}{\sqrt{1-\left(\dfrac{|\vec{v}|}{c}\right)^2}} = \dfrac{1}{\sqrt{1-\left(\dfrac{6.36 \times 10^7 \dfrac{m}{s}}{3.00 \times 10^8 \dfrac{m}{s}}\right)^2}} = \dfrac{1}{\sqrt{1-(0.212)^2}}=1.02$$ | ||
+ | |||
+ | Finally, we compute the momentum vector. | ||
+ | |||
+ | $$\vec{p} = \gamma m \vec{v} = (1.02) (9.11\times10^{-31} kg) \langle -2.05\times10^7, | ||