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Problem 49 Hard Difficulty

A proton has 1836 times the rest mass of an electron. At what
speed will an electron have the same momentum as a proton moving at 0.0100$c ?$

Answer

0.999 c

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Top Physics 103 Educators
Elyse G.

Cornell University

Marshall S.

University of Washington

Zachary M.

Hope College

Meghan M.

McMaster University

Video Transcript

So this question We have to compare the momentum of the proton and the momentum of electoral. The momentum of the proton must be equals to the moment. What electoral momento? Elektrim? Is it close to the mass of the electron times Velocity of the electron divided by the correction factor off one miners the velocity of the electron divided by the speed of light square. Now we solve this equation for the electrical Ah city. So we begin by squaring both sides of the equation. To get the momento of the proton square is because to the mast off the electron square times velocity off the electoral squared divided by one miners. The velocity of the electoral divided by C squared. Now we send this term to the side Get B B squared minus peopie squared the divided by C square. Is it close to the mass of the electron squared times The velocity of the electron squared for simplicity. Let me reel right this as the square divided by C square. Now we send this term to the other side. So Peopie square is because to m e where'd the square plus BP square B squared, divided by c squared. We can write it as P p squared being equals two V squared times and e square, plus be square divided by C squared. So now we can send this term back to the other side. To get the square is he goes to Peopie Square, divided by M e squared plus B B squared, divided by C squared. Now we take the square root off both sides of the equation. You get the is equals two be beat kinds divided by the square root off M E squared plus peopie square divided by C squared. Now all we have to do is plaguing the value off the momentum off the proton. But what is the momentum of the proton two? The momentum of the proton pp is equal to the miles off the proton kinds. The velocity of the proton divided by the correction factor off one miner's VP reminded by C squared. Asked even by the problem, V P is equal to 0.1 tank See, So BP divided by sea Easy coaster 0.0. What then we have the following BP is he goes to MP times 0.1 time see divided by the square root off one minor 0.1 It's queer. No, all we have to do it's bugging these results in the equation for the velocity of the electron. So we got the following The easy goes to MP times 0.1 time seat divided by the square root off M E Square plus P p squared divided by C Square. So these this plus m b squared times 0.1 square time sees where Divided by one minor 0.1 square under denominators Multiply it by C Square. There is a simplification off the C squared factors, and we're left with and B times 0.1 time see divided by the square root off M E squared plus empty square times 0.1 square invited by one minus 0.1 square. Now we can do the following used the order information that the problem gave us empty. He's equals to 1836 times m e. Now bringing this result in the equation you're working on to get the following the he's equals two. 1836 times m e times 0.1 Kind seat divided by the square it off M e Squared plus and square times Bureau Point. You know one where times one, eh? We think square divided by one minus 0.1 square. Now I know the following. There is a common factor off m E squared inside of the square roots so we can take it out to get the following m e. Times 1836 times 0.1 time See divided by empty times square root 01 plus 0.1 square times 1836 squared, divided by one minus 0.1 square. No, we can simplify this factors off empty and finally get the answer that the is approximately 0.999 times the speed of light.

Brazilian Center for Research in Physics
Top Physics 103 Educators
Elyse G.

Cornell University

Marshall S.

University of Washington

Zachary M.

Hope College

Meghan M.

McMaster University