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Problem 44 Easy Difficulty

Supply the details of the calculation of $\alpha$ and $\beta$ in Example 8.12$(\text { Section } 8.4) .$

Answer

$\alpha = 36.9^{\circ}$ and $\beta = 26.6^{\circ}$

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Video Transcript

following example 8.12. We know that there are two balls. Agent B ball A is bigger than and heavier than Bowlby and Ball A heats Bowlby. But that collision isn't had on the reform. Each ball will follow a different direction after that collision. Now the problem evaluates almost everything to us. But there is the last step where we need to evaluate the values off the angles. Alfa and Mita, on the example, do not provide details for that calculation. Then our task here is just serving this system off equations. Then for that, we can use the technique, for instance that is given in the example itself. Before that, let's begin by actually evaluating this multiplication toe the adequate number off significant figures. So we have the following. Let me call this equation number one and this one equation number two. So for equation number one, we have 0.500 times 4.0 Multiplying by 0.5 is the same as divided by two. So this is equals to two. But we have three significant figures. So 2.0 on these is equals to again 0.5 times True, 0.5 times two is just 1.0 times the co sign off Alfa plus 0.300 times 4 47. And these results in 1.341. Note that there is one extra digit here that we will be using throughout the calculation. But at the end we were around back to the original number of significant figures, which is three. Then this multiplies the co sign off Vita. Now, for equation number three, we have the following. Zero is equals to 0.500 times two which is 1.0 times design off Alfa minus 0.300 times 4.47. We know that this is equals to 1.341 because you had already done that calculation and these multiplies is sign off Vita. Now we have to eliminate either Alfa or Beta. As you can see, we might be tempted to just add these two equations because then we can factor the co sign off Vita the sign off Vita, the co sign off Alfa and the sign off Alfa. But this will not help us at all Because what is true is that the sine squared plus the coastline squared is equal to one, not the sign, plus the co sign instead, what you're doing is serving both off these equations for the beat A quantities. So we solve equation one for the co sign off Bitta on equation to for the sign off Vita beginning by equation one, we have the following We begin by sending this term to the other side and inverting the equation. So we have 1.341 times the co sign off bitta meaning it goes to true 0.0 minus 1.0 times. They co sign off Alfa now for equation number two, We have the following so we send this term to another site and we're done So 1.341 times that sign off Vita is equals to 1.0 times the sign off Alfa. Now we square both equations so that we have equation number one squared and this is 1.341 squared times The co signed squared off Vita This is equals to 2.0 minus 1.0 Co sign off Alfa squared on for equation number two squared. We have the following 1.341 squared times designed Squared off Vita is equals to 1.0 squared times the sine squared off Alfa. Now, as you can see, we can add equations one and two and factor this term so that we'll end up with at some between the sine squared on the Coastline square look so by doing the following one squared plus two squared We got 1.341 squared times. The sign squared off Vita plus the co sine squared off Vita is equals True, 2.0 minus 1.0 times the co sign off Alfa squared plus 1.0 times this sign squared off Alfa Then this According toe, that identity is equals to one So we end up with the following 1.341 squared is equals to 1.0 times the sine squared off Alfa Plus Now we expand this So by expanding that we got the following 2.0 squared plus 1.0 squared times. The coast sine squared off Alfa minus two times 2.0 times 1.0 times They co sign off Alfa. Now there is one thing that appears here we have 1.0 times design squared plus 1.0 squared times The coastline squared again. We can factor 1.0 to get the following one point. 341 squared is 1.0 times the sign squared off Alfa plus the code sine squared off Alfa. This is because 1.0 squared is 1.0 times 1.0 which is just itself. Then we have plus 4.0 which comes from these 2.0 square. Then we have minus 4.0 times that co sign off Alfa, As you know, this is just equals toe one. Now, to continue this question, I have to release a few parts off the board that I am not used anymore. Okay, now proceeding. We simplify this which is just equals to one to get the following 1.341 squared is equals to 1.0 plus 4.0 minus 4.0 Co sign off Alfa, then 1.241 square is 5.0 minus 4.0 Co sign off Alfa. Now we solve this equation for the co sign off Alfa. We begin by sending this term to the left hand side and this other term to the right hand side. By doing that, we get the following 4.0 times. The co sign off Alfa is equals to 5.0 minus 1.341 squared. Therefore, the co sign off Alfa is equals to five minus 1.341 squared, divided by four. Then Alfa is the inverse co sign off five minus 1.341 squared, divided by four. And the result off this calculation is approximately 36.9 degrees. The reform Alfa is a cost of 36.9 Now for Vita we plugging this result for Alfa back into this equation For that I don't need some space so let me erase a few bits off this question that I will not be using anymore. Okay, so that is plaguing Alfa into equation number two that is written like this. So 1.341 sign off vita is equals to 1.0 times they sign off 36.9 degrees. Then the sign off Mita is equals to 1.0 times the sign off 36.9 degrees, divided by 1.341. So Vita is equals to the inverse sign off 1.0 times design off 36.9 degrees, divided by 1.341 and their results in an angle vita that is approximately 26.6 degrees. And this is the answer to this question.