A proton in a cyclotron has a velocity of 2 * 10⁸ m/s, which is nearly the speed of light. The uncertainty in the velocity is 1%. What is the minimum uncertainty in the position of the proton in meters? The proton mass is 2 *10^-²⁷ kg. A) 3 *10^-¹¹ m B) 1 * 10^-²² m C) 1 *10^-¹⁸ m D) 3 *10^-⁴¹ m E) 1 *10^-¹⁷ m

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Quizplus · Accepted Answer

The uncertainty in the position of the proton is given by the Heisenberg uncertainty principle:$$\Delta x \Delta p \geq \frac{h}{4\pi}$$where $\Delta x$ is the uncertainty in the position, $\Delta p$ is the uncertainty in the momentum, and $h$ is the Planck constant.The uncertainty in the momentum is given by:$$\Delta p = m\Delta v$$where $m$ is the mass of the proton and $\Delta v$ is the uncertainty in the velocity.Substituting these expressions into the Heisenberg uncertainty principle, we get:$$\Delta x \geq \frac{h}{4\pi m\Delta v}$$Plugging in the given values, we get:$$\Delta x \geq \frac{6.626 	imes 10^{-34} 	ext{ J s}}{4\pi (1.673 	imes 10^{-27} 	ext{ kg})(2 	imes 10^8 	ext{ m/s})(0.01)}$$$$\Delta x \geq 3 	imes 10^{-11} 	ext{ m}$$Therefore, the minimum uncertainty in the position of the proton is 3 * 10^-11 m.

A Proton in a Cyclotron Has a Velocity of 2