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(III) For an elastic collision between a projectile particle of mass $m_{\mathrm{A}}$ and a target particle (at rest) of mass $m_{\mathrm{B}},$ show that the scattering angle, $\theta_{\mathrm{A}}^{\prime},$ of the projectile $(a)$ can take any value, 0 to $180^{\circ},$ for $m_{A}<m_{B},$ but $(b)$ has a maximumangle $\phi$ given by $\cos ^{2} \phi=1-\left(m_{B} / m_{A}\right)^{2}$ for $m_{A}>m_{B}$ .

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Physics 101 Mechanics

Chapter 9

Linear Momentum

Motion Along a Straight Line

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Cornell University

Rutgers, The State University of New Jersey

University of Sheffield

University of Winnipeg

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All right. So a pretty involved problem here, Uh, you have initially m a moving in the ex direction towards on B, which is at best. And then it scatters off at an angle after collision and scatters off with speed v a prime minute angle data prime and be at scares off with Phoebe Prime for that old data be prime. Um and so we're gonna use three conservation equations here. 1st 1 is the conservation of momentum in the ex direction. So this will be initially you have maybe a, um and that will equal m a v a prime Co sign data a prime The X component of that velocity plus M B V b Prime CO. Sign data be prime on DSO you have m B V B Prime co sign Theda Be prime is equal to M A tons via minus V. A prime co sign Dada a prime. And that's equations. One second. Conservation law is conservation of momentum in the wind direction. No momentum in the UAE direction that is equal to m A. Be a prime signed data a prime minus. This time M v V b prime side data be primed the vertical components and minus because obviously they are moving their of opposite vertical directions. Therefore, the equation we get here is, um m b B B prime sign They'd be prime is equal to m a V. A prime sign data a prime. And that's equation, too. Uh, and so the final conservation law is constitutional kinetic energy. So initially you have 1/2 m a p a squared. Finally, you have 1/2 m a v a prime squared plus 1/2 and be a vey vey prime squared. Uh, and so But you get from here is that Emmy Times via squared minus B a, uh, via primes squared eyes equal to M B V B Prime squared and we're gonna multiply both sides by B So m a M b times c squared minus v a prime squared C quote B squared BB Pride Square. And that's our equation. Three. Okay, so the first step in dealing with these is to square both sides of the equations of our square, both equations one and two and add them together. So what you get from there is is the following you get okay. M a Times Emmy square times V b prime squared times um, sine squared they to be prime plus co sign squared data be prompts a sine squared plus Constance, where of an angle is equal to one. So this whole thing is one, and that is equal. Thio m a v a minus m a v a prime co sign data a prime squared plus m a v a prime m A squared. Very problem squared. Uh, times sine squared data a prime. Okay, um, so we can simply fought Simplify both sides of that equation. What we get is m a b a b squared minus to m a squared times v a Ah, the a prime co sign data a prime plus m e squared be a private squared bear with me is equal to m A and B times V A squared minus view a prime square. So the m is drop out all the this one. A one Emmy term drops out from everything. Uh, when I'm a factor on and what and you may co sign data a private subject, What you get is consigned data. A prime scattering angle is 1/2 of V A over be a prime times one minus m B over. Hey plus V a prime over V eight times, 1/2 times one plus m B over and a All right, that is our equation on Soto. Answer part. And, uh, to show the theory, prime can take certain values. Ah, we note. Okay, for part A. We note that Emma is less than a and B um and so what we see here eyes that as v a prime equals do air. Um, the, uh What you get is the cancellation. You get one. Uh, you have one time 1/2 times one minus mbia of ram a plus one plus. And be over, miss. And the Obama is canceled. You have 1/2 times one. It's a co signed data A prime, Uh, because one in that case and so co sign of want is just zero. So they'd a primer zero degrees. In that case and on the other extreme, consider the case where V A prime, uh, goes to zero. And so, as it's going to zero, uh, this term this term is, um, Well, this term is dominating because the denominator gets smaller and smaller. Eso this term becomes less important. So when you just have this term, um, and M A is less than m beauty so and be over a mere That ratio is created that one. So this is a negative number, and so the most the smallest number co signed data taken, uh, Cosa coastline contain a hk is negative one right, So it will eventually approach negative but will eventually get to negative one Attn. That point, or the smallest value of CO side is but negative. One said at that point, the data a prime, uh, will be 1 80 degrees. So that is how the angles that scattering Ringle can very between zero and when he did it, Um And so in part, in part B, we have that m a is greater than a B. In this case, the arguments here don't apply. Instead, what we can do is we can take the derivative, uh, with respect to V A of, um co sign data a prime. And so this and so setting that equal to zero will give us the maximum of co signed a prime right on. So the derivative is 1/2. I'm just gonna write it down here is 1/2 minus V. Ever be a prime squared times one minus M B over M air, Uh, plus one clips one over the air, times one plus and be over a on, and so that, uh, will be set to zero. And so that would give your minimum V a prime be a prime men off that is equal to V eight times one minus m B over m A divided by one plus m b over m A, uh, to the power of half of the square root of. So plug it into, uh, plug into the co sign data a prime equation and you get And when you get is co sign. Well, this time we're gonna call it a prime fi. So you got co sign, uh, co sign Squared Fi Is it called to one minus embi over m A quantity squared and that's it.

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