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Problem 34 Easy Difficulty

An unstable particle with a mass equal to $3.34 \times 10^{-27} \mathrm{kg}$ is initially at rest. The particle decays into two fragments that fly off with velocities of 0.987$c$ and -0.868$c,$ respectively. Find the masses of the fragments. Hint: Conserve both mass-energy and momentum.


$\mathrm{m} 1=2.5119 \times 10^{-28} \mathrm{kg}$
$\mathrm{m} 2=8.8243 \mathrm{X} 10^{-28} \mathrm{kg}$


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Video Transcript

{'transcript': "in this exercise, we have a particle that has a mass on three point 34 times 10 to the minus 27 kg and is initially at breast. And then it decays into into two particles Ah, that have a mass and one and m two and have a speed. We want envy too. The one is equal to 0.987 times the speed of light, while the two is equal to minus 0.868 times the speed of light. And our goal is to find what are the masses off these two fragments? That is what is in one and two. And in order to find in one and two, apply both conservation of energy and conservation off moment. So that's briefly review how to, uh, how we deal with momentum and energy in general being in special too. So remember that the relativistic definition of momentum is gamma times. M V. We're gonna is equal to one over the square root of one minus B squared overseas square, and the energy is equal to you're more times M C square now. Conservation of momentum. Let's start with conservation off moment. Initially the particle is a breast, meaning that it has no initial momentum. So the initial momentum is zero. And the final women's um of the particles is got my one. I am one of you want. This is the momentum of the first particle plus given to m two V two. This is the moment. A lot of the second part. Okay, Now I can I'm gonna isolate m one here, so get that M one is evil too. Minus going to V two and two, The very back them a one and one going one of you one. Let's keep this is going to use with tourists and now we can go on to conservation. It would enter you to have the initial interview of the particle Noticed that the particle injuries is a breast, meaning that Gummer the initial gamma is equal to one. So the initial energy just EMC square and the final energy is equal to Goma. One I'll come one. I think it's good to find a here. Come a one is equal to one over the square root of one minus V one square, divided by C square while coming to is the same thing, but with the one changed to be tubes, we have you to square divided by C squared. Okay, so EMC squared is gonna one time Zim Once he square, let's give my to times in to see Swear so, motors that the cease cancel out so em is equal to them. A one and one. Let's give my chewing too. Okay on. And what I'm gonna do is to substitute the value that I found for m one. Uh, when we were working with conservation of momentum. So em is equal to minus and to be to go to. But if you want plus m too immature, have a m is equal to minus I'm sorry. And to two times one minus V two Overview one. So I'm to is equal to m divided by you. Don't want to Times one minus b two over you want. And this is a good time to calculate expensively Gonna one tender mature Ah, so I'm gonna just right again here on the side of the values of your one and t two. You want his 0.9 87 times the speed of light Are we to his minus 0.8 68 times the speed of light. You know one, and we can calculate it simply by inserting into the reason. One. Over the square root of one minus 0.987 square. No one is equal to 6.22 while villain too, which is one over the square root of one who, minus 0.868 square is equal to two point over. Okay, so in two is equal to M one, which is this just m The initial mass. That's 3.34 times 10 to the minus May 7 kg divided by going to which is 2.1 times one minus Ah, V two. That's minus 0.8 68 So that's why change the minus sign into the blue sign here, invited by 0.98 10. So in two is equal to 8.84 times 10 to the minus 20 kilograms and this is them to so half the year is done. We just need to have let him won't know remember that one is equal to minus and to the to them a to they had a way of you want. So in one is minus 8.84 times 10 to the minus 28. This them to times V two. That's minus 0.8 68 p. M. C. Um, Okay. And see, I'm gonna just go ahead and substituted for three times into the eighth meters per second. That time's gonna chew, which is equal to 2.1. Uh, here. There should be a good one here in formal, um, start and divided by your 0.987 times the speed of light three times into the eighth Ministry, second times 6.22. So I noticed that the speed of lichens without, and we obtain the value off one who to 2.51 time sent to the minors. 28 kg. You're So our answer is M one. It was 2.51 times into the minus eight and M two equals 8.84 times into the minus. Turn Eat worms. This is the answer to our exorcists."}

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