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Professional ApplicationErnest Rutherford (the first New zealander to be awarded the Nobel Prize in Chemistry demonstrated that nuclei were very small and dense by scattering helium-4 nuclei $\left(^{4} \mathrm{He}\right)$ from gold-197 nuclei $(197$ Au $) .$ The energy of the incoming helium nucleus was $8.00 \times 10^{-13} \mathrm{J},$ and the masses of the helium and gold nuclei were $6.68 \times 10^{-27} \mathrm{kg}$ and $3.29 \times 10^{-25} \mathrm{kg},$ respectively (note that their mass ratio is 4 to 197$) .$ (a) If a helium nucleus scatters to an angle of $120^{\circ}$ during an elastic collision with a gold nucleus, calculate the helium nucleus's final speed and the final velocity (magnitude and direction) of the gold nucleus. (b) What is the final kinetic energy of the helium nucleus?

(a) $v _ { \text {Alus } } ^ { \prime } = 5.56 \mathrm { s } \times 10 ^ { 5 } \mathrm { ms } ^ { - 1 }$ and $\theta _ { \mathrm { A } u } = - 29.48 ^ { \circ }$(b) $K E _ { \mathrm { f } _ { \mathrm { H } } } = 7.52 \times 10 ^ { - 13 } \mathrm { J }$

Physics 101 Mechanics

Chapter 8

Linear Momentum and Collisions

Moment, Impulse, and Collisions

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McMaster University

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here Festival will assume that golden nucleus is at a region and helium nucleus is mooing Along XX is in the positive direction. So this is the case before collision and this is the case after collision here. Helium nucleus is secret er at an angle of 100 20 degrees. Let the single is denoted by theater. On fire is the angle and which goal nucleus moves after scattering as we are given the kinetic energy of incoming helium nucleus, that is kinetic energy. Initial kinetic energy. A theory, um, nuclear is equal toe. I had my 00 tie in 10 to the power minus 13. You're as we know, that kinetic energy is equal toe half into mess Ophelia in tow. So get off initial ship it off helium that will be equal toe 8.0 times 10 to the power minus 13. You're as we are given. Mess off William in the problem, we can easily calculate initial See Peter Philly, um, nucleus. And that will be equal toe initial. Cpt, off William Nucleus. Equal toe one point 55 times 10 to the power seven meter par sicken. It's now in the first part of this problem. We want to calculate ship it off helium nucleus as well as we lost the off gold nuclear's. So first, we will use consideration off momentum in X direction. Now we can write mo mentum off helium nucleus before scattering. That is equal toe mess off. William Times initial sapyta Philly, um, as gold atom is at rest. So we have momentum AB goal Nicholas before secretary equal to zero and there will be equal toe momentum in next direction after scattering that would be called to mess a Philly, um, a tongue in tow component off. Sleep it off. William Adam in XY direction, which is we have atty in tow or sign after the it is, though x component off momentum off William Nucleus unless x component of momentum off gold Nicholas visually equal toe mess off gold nucleus into final Sip it off. Gold. Nicholas in tow. Co sign off Fight. Nobody rear ending we can, right. This is mess off. Mess off. Gold in tow. Final ship it off. Gold in tow. Cassina fi Is he going toe mess off. Helium nucleus in tow. Initial Sleep it off. Helium nucleus minus mesa Philly, um, Nucleus into finance. Tip it off. Helium nuclear's in tow because I know off two dogs. Nobody taking secure on both sides we can write about education is so Get off, Miss off gold in tow. So get off financing Pity off. Gold coziness secure. Fight is equal toe, so get off Mess Ophelia in tow. Sakharov initial separate off. William Balance. Secure a mess. Ophelia into Sakharov Finance. Repeat, Ophelia into Cassina. Security tha Minus to into. So get off! Miss Ophelia in tow. Final velocity. Ophelia into we'll sign off. Tita, let this in. India's equation. Yeah. Now. Well, now we'll take the conservation of momentum. In wide direction is Sir Peter Philly? Um, Nucleus is along X axis before Secretary, so it will have momentum in wider. He will be called zero plus goal. Nucleus is the arrest. So it will have momentum equals zero. That will be equal toe momentum off helium nucleus after securing in wide direction, that is mess off. Helium in tow. Final. See Peter Philly, um, into sign after dark for us. Mess off gold in tow. Finance. Tip it off. Gold in tow. Sign off. Fight. Nobody ranging we can. Right, This is mess off. Gold into final Sip it off. Go on, sign off. I is equal Toe minus mess off. William in tow. Finance sequitur! Philly um, in tow. Sign off theater. Let this nm There's equation one again by taking so get off equation we can write, So get off myself Gold in tow. So get off final sip it off Gold into sign off. Secure Fi is equal toe mess off. William secured in tow. Secure a final sapyta Philly, um in tow. Science care Tita, Let this is, um this equation me Now we can at equation and be we can write equation and be after Eddie is so Kyra Mess off Gold in tow. Sakharov Final spit off gold in tow because science secure fibulas sign secure if I equal to so get off Miss Ophelia in tow. So get off initial sapyta Filion Plus So get off mess Ophelia in tow. So get off finances pedophilia in tow or sign up to care data minus twice off. So Karamazov ilium in tow. Initial separate. Er, Philly. Um in tow. Final step iturf William into prison after dark. Bless. So get off Mess Ophelia in tow. So get off. Final sequitur. Philly, um, in tow. Sign off, Superior Tita. No! By using till noma tree, we can write because science security Topless. So, Rico, Science Soccer five plus science to clarify equal toe one. So this will. This will eliminate the fight from from the equation. Now we can, Right? This is so Kyra. Mass off gold in tow. Sakharov Finance, Ship it off. Goal is equal toe, so get off Mess, Ophelia. Andi in tow. Sakharov Initial Superior. Ophelia, you're in the second time on. Right inside in the last rung on right hand side. Also having for science security guard Paula Science Aquatica and again Because science, you get it up. Alaska. Science. You get it? I will be equal to one. This gives us mess off. William secured in tow. Final supporter. Philly. Um scared. Minus twice off. Miss Ophelia secured in tow. Initial supporter for William in tow. Final separate. Ophelia in tow. Kusenov. Kita. No way. Rearranging we can. Right, This is final. Ship it off. Gold Suker is equal toe. Miss A Philly, um secured, Divided by mess off. Gold secured in tow. Sakharov initial supporter Ophelia for less. So get off my recipe. Turf, William. Minus do in tow. Initial Spitter. Philly. Um, in tow. Final step iTurf William in tow. We'll sign off Kita that this is this numbers equation. See, as we are given in the problem, that collision is elastic so we can use conservation off kinetic energy here, that is initial kinetic energy that is kinetically before scattering. Must have equal toe kinetic energy after scattering so kinetic energy before securing is equal toe half in tow. Mess Ophelia in tow. Initial. See Peter Philly. Um, plus initial kinetic energy of gold necklace zero. They're very equal toe final Kinetic energy off William, which is half in tow meself. Ilium in tow. So get off Final Supporter, Philly. Um well, that's half into mess off gold Nuclear Scinto So get off final Sleep it off. God! Here we will cancel half from both sides of the equation. This gives us so get off finance. Tip it off. Gold Equal toe, Equal toe mess off. Union Nuclear's divided by myself. Gold into So get off initial. See Peter Ophelia minus Sakharov Final See Peter Ophelia. Let this is in them. There's equation Geep Now we have two equations. That is the equation, See? And recording d both having the trunk. So again, we have goal our left hand sides. Now we can easily equate these two equations by equating you couldn't see and the we can write Mess affiliate, um squared, divided by meself l ammonium secured into initial See Peter fill Liam's occurred with us financier Peter Philly. Um secured. Managed to into initial, sir Peter Philly, um, into final see Peter Ophelia into cozy NAFTA is equal toe mess Ophelia divided by mess up in ammonia into initial she Peter Philly, um secured minus final. Secure off. William secured by rearranging we can, right. This is Miss a Philly. Um, you added by mess off gold the last one in tow. So get off. Fantasy pedophilia minus two into initial, Sir Peter Philly, um, into Susantika in tow. Massa Philly, um divided by a mess of gold in tow. Final Sleep it off. Helium balance. Mess off, William divided by a mess of gold minus one into initial, Sir Peter off. William Sucker. That is equal to zero. So this is a quadratic equation in the variable. We have helium, so here. This is the trump. Yeah, and this is the trunk the and this is the constant room. See, now we can simplify this equation by putting different values. Yes. The ratio between Miss Ophelia on mess of goal is equal toe zero pied 0203 Last one into we have helium secured minus two in tow. We have already calculated at the start of problem that initial tip it off. Helium nucleus is 1.55 times 10 to the power seven meter parsigian in tow. Because I nafta is angle at which helium nucleus skater is 120 degree. And again, the ratio between the mess off helium and massive goal is zero fied 02 zero tree in tow. We have Hey, Liam. Unless zero pied 0 to 03 minus one into 1.55 times 10 to the power 7 m Parsigian secured is equal toe one. By simplifying this accordion, it gives us one point 02 geometry, so get off final See Peter Ophelia minus 31 for 65 0 times. Finance. Sip it off. William minus two, Part 35 times 10 to the power 14 equal Tau zero. So this is a quadratic equation here. I have neglected inner city with the constant strong as well as with the confessions. Now we can use quadratic equation to solve for fine, insipid off. William, we will get to answers. One will be positive and other will be negative as magnitude off support is always positive. So we will take the positive value on a positive value will be equal toe one point five times 10 to the power seven meter Parseghian. Now we can put this value in equation ik would indeed by putting this value in equation d, we will get final Sip it off gold Nicholas and there will comes out fire part five seats times 10 to the power five Major Parsigian. So this is the magnitude off nebulous tree off golden nucleus After Secretary Now we will find a direction off Golden Nicholas after secretary again. There waas equation which Waas named the equation one by putting different values in Equation one. We will get fight equation one will become miss off gold in tow Finance Ship it off Gold sign off. I is equal toe minus mess off. Helium in tow. finances. Peter Philly. Um, in tow. Sign off, Tita. Here myself. Goal is given final. Sip it off. Gold we have already calculated, which is 1.5 times 10 to the power. 7 m per second myself, freedom is given. We heard calculated the final speed off. You're young and its allies equal toe 120. By putting all these values, we will get five. That is the angle at which gold nucleus see critters and there will be equal toe minus 20. Empire she degree, that is the golden down will scatter at an angle of 28.3 degrees counterclockwise from XX is that is 28.3 degrees from the initial direction off you young nucleus. Now in the last part, we want to calculate final kinetic energy off helium nucleus by using the relation final kinetic energy are feeling nucleus that is equal toe off times. Mess off, William Nuclear Times Superior off. Sleep it off. Secure our finances. Pedophilia By putting different values, we will get final kinetic energy. We'll be cool Toe 7.52 times 10 to the power minus 13 June. There's the end of the problem. Thanks for watching

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