Two protons, starting several meters apart, are aimed directly at each other with speeds of 2.00x10^5 m/s, measured relative to the earth. Find the maximum electric force that these protons will exert on each other.

Video transcript

Hey everyone. This problem is working with cool um slaw, Let's see what it is asking us. We have a linear particle accelerator accelerating to hydrogen ions at equal speeds towards each other. We are given a speed at the point of infinite separation and we are asked to determine the greatest electric force that the two ions will exert on each other. So the clue here electric force um two ions exerting on each other. That should clue you in right away that we were working with Poland's law. So let's recall that that equation for electric force is K. Q one Q two over R squared. Where K is cool is constant. We recall that that is 8.99 times 10 to the nine newton meters squared over M squared. Uh Q one and Q two are the same because the two particles are both hydrogen ions. And we can recall that a hydrogen ion has the same charge as a proton, which we recall is 1.6 times To the -19 columns. Okay, But what we don't have here is distance. We do know however, that it's a closed system. We have constant energy in the system. So that gives us a hint that we can use our conservation of energy. So we're working with two ions here. So our conservation of energy will be two K not plus you not equals to K. F. Plus you, F or K. Is the kinetic energy. And you as the potential energy. We know we don't have any potential energy initially because we have infinite separation. So that term goes to zero. And we know that we don't have any kinetic energy at the final point when the two ions will hit into each other, there will be an instantaneous moment of rest there. And so that turn goes to zero. So our conservation of energy can simplify to two K. Not equals U. F. Let's recall. Our kinetic energy is given as one half and V squared. And potential energy for um charged particles is given as K. Q. One Q two over. R. That's very similar to columns law for the force. But now we have a now we have speed which we were given in the problem. And so the only unknown in this second equation is our So let's just make a note here that speed at infinite separation. That's our Um the not term is 6.2 times 10 to the 5th m/s. Oh and um mass of a proton is also a constant. Let's recall. That's 1.67 times 10 to the - kg. So what I'm going to do now is um rearrange this equation in terms of our so what that gets us these cancel, we know that the queues are the same. So that becomes R equals K. Q squared over and B squared. Now we can plug that into our columns law for this unknown are here and columns law becomes que que and we read that it's Q squared, right? Because the queues are the same. The charges are the same. Um Over our squared. So one over R. Is um be not squared over K. Q squared. And that are is squared. So I'm just gonna write out the term again. This is just kind of to make it easier for us to see what cancels and how to simplify. So the electric force is N. Squared. The not the four over K. Q squared. So from here we can finally plug in our values that we have. All of these are known. And we'll get into our calculators and get an answer for force. So we have the mass 1.67 10 to the minus 27 kg. That quantity squared. Our speed was 6.2 times 10 to the fifth meters per second. Power for uh Poland's constant 8.99 times 10 to the ninth. Newton meter squared per Pullum squared. And our charge 1. times 10 to the minus 19th columns swear That comes out to 1.79 times 10 to the -3 mutants. Alright, so let's take a look at our potential answers and it looks like the correct answer here is going to be c 1.78 times 10 to the minus three mutants. All right, that's all we have for this problem. We'll see you in the next video