A neon atom $(m = 20.0 \textrm{ u})$ makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a $55.6 ^\circ$ angle from its original direction and the unknown atom travels away at a $-50.0 ^\circ$ angle. What is the mass (in u) of the unknown atom? [Hint: You could use the law of sines.]

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. This neon particle comes in and collides with another particle that's initially at rest and because the collision is elastic that means the total kinetic energy before equals the total kinetic energy after the collision. So beforehand, we have all the kinetic energies just in this neon particle so we have m 1v 1 squared before the collision— I really didn't bother writing the one-half before everything because it's just a common factor that would cancel anyway— and then after the collision, we have kinetic energy of particle one is m 1v 1 prime squared where v 1 prime is its speed after colliding and we have kinetic energy for particle two is m 2v 2 prime squared. So that's all we can say as far as energy is concerned and then we can talk about momentum but the question gives us a tip to use the law of sines and that means we can say that the vector momentum of particle one after the collision where I have copied this arrow here in green plus the momentum of particle two and when I say plus, I mean the vector sums so we are adding these arrows together equals the total momentum we had beforehand. And you can always draw a triangle like this for all your momentum questions but it happens that in this question because we are given two angles and no velocities that drawing a triangle is the best strategy for answering it. So the law of sines says that the length of any side in this triangle divided by the sin of the opposite angle equals the length of any other sin divided by the sin of its opposite angle. So m 1v 1 divided by the sin of this angle α equals m 2v 2 prime divided by the sin of the angle opposite it which is Θ 1 and that equals m 1v 1 prime divided by the sin of the angle opposite it which is Θ 2 because this dotted line is along the x-axis and along the parallel to the direction of the neon's initial velocity and Θ 2 is this angle below the horizontal for the particle two after the collision and we have interior alternate angles here; this is Θ 2 where here is one parallel line, here's another parallel line and here's the transverse between them and this angle and this angle are interior opposite angles so they are equal so that's how we get Θ 2 inside the triangle there. α then is gonna be 180 degrees, the sum of all angles in any triangle minus the other two angles this one being Θ 1 and this one being Θ 2 so there we go. And we are gonna solve for v 1 prime and then solve for v 2 prime in terms of m 1 and v 1 and then we'll substitute into this energy formula for v 1 prime and v 2 prime and we'll have an equation only consisting of m 1 and our known angles. So v 1 prime that comes from looking at this term and this one and you can multiply both sides by sin Θ 2 and divide both sides by m 1 and we are just ignoring this thing here, we are just imagining an equation that looks like this m 1v 1 prime over sin Θ 2 is m 1v 1 over sin α— you can match any pairs here in this sequence of equalities if you like. So rearrange that for v 1 prime and you get m 1v 1 sin Θ 2 over m 1 sin α and the m 1's cancel giving us v 1 sin Θ 2 over sin α and that will be useful for substitution later. And then we solve for v 2 prime from here and it's gonna be m 1v 1 times sin Θ 1 over m 2 sin α. And then substitute for each of these in our energy formula. So I have copied the energy formula here with substitution's for v 1 prime and v 2 prime and so that's m 1v 1 squared equals m 1 times v 1 sin Θ 2 over sin α squared plus m 2 times v 2 prime where I have written m 1v 1 sin Θ 1 over m 2 sin α instead of v 2 prime and that gets squared. And then we carry out the squaring here and that gives us m 1v 1 squared sin squared Θ 2 over sin squared α and here's m 2 canceling with one of the m 2's in the bottom here giving us m 2 to the power of 1 in the bottom here and we are left with m 1 squared v 1 squared on top sin squared Θ 1 over sin squared α. Then divide every term by m 1v 1 squared so the m 1v 1 squared cancels everywhere and this becomes m 1 to the power of 1 so that m 1 still stays there and then multiply everything... well, let's do it like this; we are gonna take everything and multiply it by m 2 sin squared α over m 1v 1 squared. So I just talked about the dividing everything by m 1v 1 squared and then let's carry on with multiplying by m 2 sin squared α. So the m 2 sin squared α will disappear from this term and we are left only with m 1 to the power of 1 times sin squared Θ 1 and then on this term here, the m 1v 1 squared cancels away, the sin squared α also cancels away leaving us with sin squared Θ 2 multiply by this m 2 here. And then in this term, the m 1v 1 squared divided by m 1v 1 squared makes 1 and then multiply it by m 2 sin squared α. And then we'll take this term to the left side by subtracting it from both sides and factoring out the m 2 so m 2 times sin squared α minus sin squared Θ 2 equals m 1 sin squared Θ 1. Then divide both sides by this bracket to solve for m 2. And there we have it; m 2 is m 1 times sin squared Θ 1 over sin squared α minus sin squared Θ 2 and we can plug into our calculator now. So we have m 2 is gonna be the 20.0 atomic units of the neon nucleus times sin squared of the angle of deflection of the neon which is 55.6 degrees above the x-axis divided by the sin squared of 180 minus 55.6 minus 50 and then minus sin squared of 50.0 degrees and that gives 39.9 atomic mass units. Now notice how you plug in sin squared of something into your calculator; make sure you type sin of some angle and then close the bracket and then type the square; if you did sin of 55.6 squared, that would be bad because you are taking the sin of the number squared but instead what you want is the sin of the number and then square the result of that sin. Hope that makes any sense. So it has to be written this way with closing the bracket on the angle and then squaring.