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In Conceptual Example 8.1 (Section 8.1 ), show that the iceboat with mass $2 \mathrm{~m}$ has $\sqrt{2}$ times as much momentum at the finish line as does the iceboat with mass $m$.

$\sqrt{2} p_{A}$

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University of Michigan - Ann Arbor

Simon Fraser University

Hope College

McMaster University

{'transcript': "Our question says to show that the kinetic energy K of a particle of mass M is related to the momentum P is equal to the square root of K squared plus two K M C squared. See? Okay, well, we're gonna go ahead and use equation 36 deaths 11 and 36. That's 13 of the 1st 1 says E is equal to que plus m c squared. So the total energy is equal to the kinetic energy. Plus the rest mass energy and PC squared is equal to e squared minus M c squared. Squared. Okay, well, let's plug in our value for E um, on the left side of the equation when I just marked by a star here into here, we're just pointed with zero. So we have PC squared is equal to okay plus M c squared, which is our expression for the total energy that's all squared minus M. C squared squared. Okay, so if you carry out the square operation there for K plus M C squared, that's equal to K squared plus to K. M. C squared and then plus M c squared squared. But then we also have a minus m c squared squared. So those two terms, we're just gonna cancel So we could just leave it as this case Square plus two k and C squared and we want to solve for P. So we're gonna have P is equal to first. We'll take the square root of both sides a square, root this side right, and then you square it this side and then we'll divide, uh both sides by sea. So you'll end up with the square root of K swear plus to K. M. C squared, and all of that is divided by sea. And that was the expression we were asked to show so we can box it in. That's our solution."}