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Problem 3 Medium Difficulty

(a) Show that the kinetic energy $K$ and the momentum magnitude $p$ of a particle with mass $m$ are related by $K=p^{2} / 2 m .$ (b) A $0.040-\mathrm{kg}$ cardinal (Richmondena cardinalis) and a $0.145-\mathrm{kg}$ baseball have the same kinetic energy. Which has the greater magnitude of momentum? What is the ratio of the cardinal's magnitude of momentum to the bascball's? A $700-N$ man and a $450-N$ woman have the same momentum. Who has the greater kinetic energy? What is the ratio of the man's kinetic energy to that of the woman?

Answer

(a) $\frac{p^{2}}{2 m}$
(b) $1.90 p_{c}$. The baseball has the greater magnitude of momentum. $p_{c} / p_{b}=0.526$
(c) $1.56 K_{\mathrm{m}}$. The woman has greater kinetic energy. $K_{m} / K_{w}=0.641$

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Video Transcript

in the first item off this question, we have to show that the kinetic energy as a function off the momentum is given by this expression. To do that, we begin by solving this equation for V by doing that, to get the following V is equals to P divided by M. Now we substitute this equation for V into the equation for K. The result is the following key is given by the mass times V, which is p divided by M squared, divided by troop. And this is m times B squared, divided by two times m squared. We can simplify out when factor off em like this and then we get our answer so okay can be given by p squared divided by two times m. In the second item, there is a board with a mass off 0.4 kg that is flying to the right with velocity V c and a baseball with a mass off 0.145 kg that is also flying to the right with velocity V b. We know that birth off these items have the same kinetic energy. Then we have to define which one has the greater momentum for that. We can use these expression from the previous item, so we know that both kinetic energies are the same. This means that P. C squared that is the magnitude off. The momentum for the board squared divided by two times the mass off the bird is equals to the momentum off the baseball squared, divided by two times the mass off the baseball. Then we can divide both sides off this equation by the momentum off the baseball squared. By doing that, we get the following B C squared, divided by two times m C. Times B B squared is equals to one divided by two times the mass off the baseball. Then we can send this term to the other side, multiplying one by doing that to get the following P C squared, divided by P B squared is three times the mass off the cardinal divided by three times the mass off the baseball. We can simplify out these factors off true, and finally we can get the ratio off. The momentum's so PC, divided by PB, is given by the square it off the mass off the bird divided by the mass off the baseball. We know that the mass off the bird is much smaller than the mass off the baseball, the reform that racial is smaller than one. Which means that the momentum off the baseball is bigger than the momentum off the word. So PC is smaller than PB. Now we have to evaluate the racial to evaluate the racial. How we have to do is splitting the masses. So this is the square it off 0.4 divided by 0.145. This is 0 40 on these results in approximately 0.5 to 6. And this is the answer to the second item off this question. For the next item, I need some space. So I you raised the answer for item B. In the third item, we have a man that waits 700 Newtons and a woman that waits 450 Newtons. They have the same momentum. They're moving, say to the right. So just like that, now we have to get your mind who have the biggest kinetic energy. In order to do that, we can again use this equation so we can solve that equation for P on. By doing that, we get the following serving. This equation for P results in two times m times k is equal to p squared. Therefore, P is equal to the square root off two times m times k Then we know that the momentum off the woman is equal to the momentum off the men. Therefore, the square it off three times the mass off the man Let me call this m m times the kinetic energy off the man K m is equals to the square root off three times the mass off the woman and w times the kinetic energy off the woman kw. Now we can square both sides off this equation so that you get the following two times the mass off the men times The kinetic energy of the man is the cost to two times the mass off the woman times the kinetic energy off the woman. Now we can send the kinetic energy off the woman to the other side dividing And by doing that to get the following three times the mass off the men times the kinetic energy off the men divided by the kinetic energy off the woman is equal to three times the mass off the woman. Then we can simplify the factory off. True, that is appearing here. And finally send the mass of the woman to the other side. Dividing by. Doing that to get the following K m divided by K W. Is equal to the mass off the woman divided by the mass off the man. Finally, we don't know the masses, but we only know the weights since we know that the weight is given by the mass times. The acceleration of gravity and acceleration of gravity is the same for both of them. It's also true that the ratio off the weights say W m divided by W. W, is given by the mass off the man Times G, divided by the mass off the woman times G so that gravity is simplified out and the ratio between the weights is because to the ratio in between the masses, therefore, K m, divided by K W, is equals to the weight off the woman which is 450 Newtons divided by the weight off the men which is 700 Newtons. Then we can simplify the zeros and conclude that the result off this calculation. So the ratio between the kinetic energy off the man and magnetic energy off the woman is approximately 0.643 And this is actually the answer for this item off this question. So 0.643 and then by taking a look at this result, we can see that the kinetic energy off the woman is bigger than the kinetic energy off the man. So we have this relation, and it is the answer to this question.