The nucleus of the polonium isotope ²¹⁴Po (mass 214 u) is radioactive and decays by emitting an alpha particle (a helium nucleus with mass 4 u). Laboratory experiments measure the speed of the alpha particle to be 1.92×10⁷ m/s . Assuming the polonium nucleus was initially at rest, what is the recoil speed of the nucleus that remains after the decay?

Video transcript

Hey, everyone. So this problem is dealing with conservation of momentum. Let's see what it stands. Proton radioactivity is an unusual type of radioactive decay that occurs when a proton is ejected from a nucleus. During the nuclear experiment, a proton of mass one U emission was observed from the decay of Cobalt 53 of mass 53 U. The proton speed in the laboratory frame was measured to be 1. times 10 to the seven m per second by the recoil speed of the daughter nucleus F E or iron 52. Our multiple choice answers here are a 2. times 10 to the five m per second. B 7.23 times 10 to the six m per second. C 1.7 times 10 to the seven m per second or D 1.42 times 10 to the seven m per second. OK. So the first thing I'm going to do here is draw out this decay that's happening. So it helps us visualize what's going on in this crop. So we have this Cobalt atom that is going to decay you iron plus a proton So our conservation of momentum equation tells us that our initial momentum is equal to our final momentum for all of the objects in this scenario or in the system. So initially, and we can also recall that momentum is given by mass most I. So initially, we have this Cobalt atom. So we'll have mass of the Cobalt and velocity the and then after the decay, we have mass of the iron and velocity of the iron plus mass of the proton and velocity of the proton, the Cobalt atom is initially at rest. So V C is equal to zero. And that whole left-hand side of the equation goes to zero. And we are asked to solve for the speed of the iron nucleus or V I, we are looking for the magnitude of the speed. So we can ignore the the negative sign here. So V some I is equal to mass of the proton multiplied by velocity of the proton divided by mass of the iron nucleus. So from the problem, they tell us mass of the proton is one U. The speed of the proton was given as 1. times 10 to 7 m per second. And then mass of the daughter nucleus, the daughter iron nucleus is 52 U. And so when we plug that into our calculator, we get a speed of that daughter iron nucleus of sorry 2.73 times 10 to the fifth meters per second. And so that is the answer to this problem. And when we look at our multiple choice solutions that aligns with answer choice A, so that's all we have for this one, we'll see you in the next video.