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An electron and a 0.0200 -kg bullet each have a velocity of magnitude $500 \mathrm{m} / \mathrm{s},$ accurate to within $0.0100 \% .$ Within what lower limit could we determine the position of each

object along the direction of the velocity?

1.16 $\mathrm{mm}$

$5.28 \times 10^{-32} \mathrm{m}$

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Cornell University

Numerade Educator

Hope College

McMaster University

So this exercise we have an electric and a bullet whose mass is 0.2 kilograms and they both travel at a speed of 500 muted for second. And the uncertainty of the speed is uncertainty. Um, 0.1%. Okay. And we want to find what's the lower limit for the uncertainty in the position off each one off of these objects. The electric and the bullet. Okay, First of all, let's calculate boats to the uncertainty in the velocity based on the fact that the uncertainty 0.1%. Um, So this is all we have to do is to multiply 0.1 Time to stand to the minus two by 500 meters per second. Okay. And this is 0.5 meters per second. Okay, this is 0.1% of 500. Okay, this is Delta V for both Ah, the electron and the bullet. Now let's calculate what is the minimum value for their tax. So for the election, we have that the mass of the electron is given by 9.1 times 10 to the minus 31 kilograms. This is gonna be import. And from the uncertainty principle we have the Delta P Delta X is greater or equal than H bar over two. That's a B is the uncertainty in the momentum Bell tax is the uncertainty in the position and the uncertainty in the momentum is and e Delta V and this is gonna be multiplied by doubt. Acts. So this is we greater or equal then h marvel to for the reason daughter acts is greater or equal than h bar over to m e the V so it can calculate, uh, what's the lower limit for no tax by calculating Okay, h over four pi m e Delta V. So this is 6.63 times tend to the mind a study for jewel second, divided by four pi times the mass of the electorate 9.1 time Stand to the minus 31 kilograms times the uncertainty in the speed which is five times 10 to the minus two meters per second and this result in five very 1.15 times 10 to the minus three meters. Do you? This is the so we have that the attacks is greater or equal, then 1.16 times sent to the minus three. So the lower the lower bound, the lower limit for doubt. Tax for the electron is 1.16 millimeters. Okay, so Lauren limit is 1.16 millimeters, and I can do the same thing for the bullet. So here the bullet, uh, we have the mass of the bullet. I'm gonna just write it down here again, that 0.2 kilograms. And as before, just look at this equation here. The tax is greater or equal than H bar over two times the mass times out of E. So again, we're gonna have the tax is greater or equal than H bar over two and Delta V. So this is a show over for pie, and that's a V. And against substituting the numbers 6.63 times, 10 to the minus 34. You second divided by four pi times in which is two times since the minus two kilograms times Delta view, which is five times since the minus two meters per second. So this is ah, 5.28 times 10 to the minus 32 meters. Okay, so what we have is that the tax is greater or equal. Let me write this a little better. It's greater or equal, then 5.28 times since the minus 32. And that is the lower limit for the uncertainty. So here have at the lower limit is 5.28 times 10 to the minus 32 meters and this concludes our exercise.

Universidade de Sao Paulo