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183_notes:examples:momentumfast [2014/07/10 14:09] – caballero | 183_notes:examples:momentumfast [2014/07/10 14:20] – caballero | ||
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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, | + | An electron is observed to be moving with a velocity of $\langle |
==== 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, | + | * It has a velocity of $\langle |
* This velocity is near the speed of light ($c = 3.00\times10^8 \dfrac{m}{s}$). | * This velocity is near the speed of light ($c = 3.00\times10^8 \dfrac{m}{s}$). | ||
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First, we compute the speed of the electron. | First, we compute the speed of the electron. | ||
- | $$ |\vec{v}| = \sqrt{v_x^2+v_y^2+v_z^2} = \sqrt{(2.05\times10^7)^2+(6.02\times10^7)^2+(0)^2} | + | $$|\vec{v}| = \sqrt{v_x^2+v_y^2+v_z^2} = \sqrt{(-2.05\times10^7 |
+ | |||
+ | 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, |