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A body of mass 500 grams is moving at a constant velocity π£ equals two π’ minus three π£ centimeters per second, where π’ and π£ are two perpendicular unit vectors.
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Find its kinetic energy.
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So, first of all, what we can see is that weβre trying to find the kinetic energy.
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So letβs remind ourselves of the formula for kinetic energy.
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Well, we know that kinetic energy is equal to a half ππ£ squared, where π is the mass and π£ is the velocity.
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But what we want to do is consider our units.
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Well, if the units for mass were kilograms and units for velocity were meters per second, then this means that our kinetic energy would be in joules.
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However, if our mass was in grams and our velocity was in centimeters per second, then our kinetic energy is measured in ergs.
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Well, letβs take a look at our question to see what our units are going to be.
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Well, we can see that the mass is in grams and the velocity is in centimeters per second.
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So we know that our answer is going to be in ergs.
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So now what we want to do is substitute in our values into the formula to find our kinetic energy.
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So as we already said, our mass is 500 grams and our velocity is equal to two π’ minus three π£ centimeters per second.
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However, we canβt just pop these values into our formula.
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And thatβs because what weβve got here is our velocity in vector form.
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And what we need to do is find the magnitude.
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Well, if we want to find the magnitude of a vector, if you think of our vector as ππ’ plus ππ£, then this is gonna be equal to the square root of π squared plus π squared.
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And this comes from the Pythagorean theorem.
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And we can use that because we know that π’ and π£ are two perpendicular unit vectors.
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So in that case, they are at right angles to each other.
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So therefore, we can adapt our Pythagorean theorem to give us this result for our magnitude.
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Okay, so what we want to do now is find the magnitude of two π’ minus three π£.
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So therefore, we can say that the velocity is going to be equal to the square root of two squared plus negative three squared.
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So therefore, we can say that the velocity is gonna be equal to root 13 centimeters per second.
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And weβll keep it in this form because actually we want to maintain the accuracy for the next part of the question.
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And as itβs magnitude, weβre only interested in the positive result.
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So then we can substitute our values into the kinetic energy formula.
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When we do that, we get kinetic energy is equal to a half multiplied by 500 multiplied by root 13 squared, which is gonna be equal to 250 multiplied by 13.
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Thatβs because root 13 squared is just 13.
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So therefore, we can say that if a body of mass 500 grams is moving at a constant velocity π£ equals two π’ minus three π£ centimeters per second, where π’ and π£ are two perpendicular unit vectors, then the kinetic energy is going to be equal to 3,250 ergs.