What is the escape speed of an electron launched from the surface...

Question

What is the escape speed of an electron launched from the surface of a 1.0-cm-diameter glass sphere that has been charged to 10 nC?

Accepted Answer

Hey, everyone in this problem, we're asked to calculate the escape speed of a fluid ion launched from contact with the surface of a charged ball of radius 0.23 centimeters and total charge eight nano colons used to shoot charge objects into space. We're told that we can ignore the mass of electrons in the ion. We have four answer choices all in meters per second. Option. A 5.9 option B 5.6 multiplied by 10 to the exponent four. Option C 1.0 multiplied by 10 to the exponent seven. And option D 5.6 multiplied by 10 to the exponent. So let's get started. I want to think about the escape speed. When we're thinking about the escape speed, we want to think about the conservation of mechanical energy. OK. So we know in this case that we have some kinetic energy at some point because this fluorine ion is being launched, it's gonna have speed. That means kinetic energy, we have some potential energy because we're dealing with charges, right. So if we think about our conservation of mechanical energy, we can say the initial kinetic or the initial mechanical energy, emac not will be equal to the final mechanical energy. Em me now what makes up those mechanical energies we just kind of mentioned, OK. We have the kinetic energy and we have the potential energy. So knot plus U kno will be equal to KF plus UF. OK. So K is our kinetic energy, U is our potential energy. And we're just using subscript knot to indicate that initial point and f to indicate that final point when we have that escapes, I want to think about the escape speed. I recall the condition. OK. So the conditions are that we're gonna take this final distance, final radius or the final distance between these charges to go to infinity. Hey, this is getting further and further away. OK. So the fluorine ion from our charge ball, the distance is going off to infinity which tells us that our potential energy is going to go to zero. Yeah, they're gonna get infinitely far apart. So there will be no potential energy. Those charges are not going to be um creating any potential energy between each other. Now, the other thing we can think about at that point is that VF OK. That final speed will also be equal to zero. OK. Because we're gonna be at our maximum height, we're gonna be at that maximum distance and momentarily that ion is going to come to rest before switching directions. OK. So if our final potential energy is zero and our final speed is zero, meaning our final kinetic energy is zero, we have the entire right hand side being equal to zero. And for the left hand side, we have that kinetic energy. So it's gonna be one half multiplied by the mass multiplied by the initial velocity V not squared plus the potential energy. OK. Between these two charges, we can consider them point charges. And recall that that equation is going to be one divided by four pi absolutely multiplied by the charge you. And that's gonna be the charge of our ball multiplied by the charge. Qf So the charge of the fluorine ion divided by are not the initial distance between OK. And this is all going to equal zero. Now, these values, we know the only value we don't know is the not and this is actually going to be our escape. And because we've set up our conditions, the way we have this turns into the escape. OK? So that is the value we're looking for. Let's go ahead and substitute in the other values. So the mass of fluorine is going to be 19 U maybe we can convert that into kilograms. So we're gonna have one half multiplied by 19 multiplied by 1.67 multiplied by 10 to the exponent negative 27 kg and multiply by being squared again, that value we're looking for. So we can add the what constant nine multiplied by 10 to the exponent nine newton meters squared per P squared multiplied for the charge of our ball eight nano coulombs. So eight multiplied by 10, the exponent negative nine coulombs multiplied by the charge of this fluorine ion. And it's gonna have the same charge as an electron. So negative one multiplied by 1.6 multiplied by 10 to the exponent negative 19 coulombs. And this is all going to be divided by are not. So the initial distance between our flu ion and the surface of our charge phone, no, if we think about our charge ball and our fluorine ion is launched from contact with the surface and our ball has a radius of 0.23 m. OK? So the initial distance between the fluorine ion and our charged ball, ok? That's gonna be the distance from center to center is going to be the radius of that charge ball. So that is 0.23 m or we can write this as 0.0023 m. Sorry, that's 0.23 centimeters. So we can write that as 0.0023. And so we can use that in our equation. So we're dividing by 0.0023 m. Ok? So let's go ahead and simplify. This is all equal to zero. We're gonna move the constant terms to the right hand side. If we simplify on the left, we get 1.5865 multiplied by 10 to the exponent negative 26 kg multiplied by the escape speed squared. That's going to be equal to. Now we have to subtract this big constant term to move it to the right. But we have a negative there as well. Subtracting a negative is gonna give us a positive. And so we end up with 5.00 869565 multiplied by 10 to the exponent negative 15 Newton meters. We're gonna divide both sides by what we have this 1.5865 multiplied by 10 to the exponent negative 26 kg. And now we're gonna take the square root and we get that this escape speed. The escape is gonna be equal to about 5.61 878 multiplied by 10 to the exponent 5 m per second. So that is the escape speed of this flooring atom or flooring. I, I'm sorry that we were looking, if we go back to our answer choices and compare what we found. We can see that the correct answer choice is going to be option D 5.6 multiplied by 10 to the exponent 5 m per second. Thanks everyone for watching. I hope this video helps see you in the next one.

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What is the escape speed of an electron launched from the surface of a 1.0-cm-diameter glass sphere that has been charged to 10 nC?

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