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A car of mass 1350 kilograms moves in a straight line such that, at time ๐ก seconds, its displacement from a fixed point on the line is given by ๐ equals six ๐ก squared minus three ๐ก plus four meters.
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Find the magnitude of the carโs momentum at ๐ก equals three seconds.
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Weโre looking for the magnitude of the carโs momentum.
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Momentum is a course of vector quantity.
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But weโre only interested in the magnitude of this quantity.
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Now you might know that the momentum of an object is the product of its mass and velocity.
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And this is good news because we know the mass of the car.
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Itโs 1350 kilograms.
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But weโre not explicitly told the velocity of the car anywhere in the question.
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What we are told is its displacement ๐ equals six ๐ก squared minus three ๐ก plus four meters, where ๐ก is the time in seconds.
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And we can use this displacement to find the velocity as the velocity at any given point in time is just the instantaneous rate of change d by d๐ก of the displacement at that point in time.
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So we can call the velocity ๐ฃ and displacement ๐ as it is in the question.
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And we can substitute the expression we have for ๐ in terms of ๐ก.
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Weโre told that the displacement ๐ is six ๐ก squared minus three ๐ก plus four.
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And we can differentiate term by term.
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The derivative of ๐ก squared with respect to ๐ก is two ๐ก.
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And so the derivative of six ๐ก squared is six times two ๐ก, which is 12๐ก.
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The derivative of ๐ก with respect to ๐ก is just one.
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And so the derivative of three ๐ก with respect to ๐ก is three.
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And the derivative of a constant is zero.
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And so the constant term four doesnโt contribute anything to the velocity.
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We found therefore that the velocity at time ๐ก is 12๐ก minus three.
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And as the displacement was measured in metres and the time in seconds, this velocity has units of metres per second.
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So now that we have the velocity at time ๐ก, we can substitute it in to our equation for momentum.
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The momentum is 1350 times 12๐ก minus three.
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And as the mass was measured in kilograms and the velocity was measured in metres per second, this momentum is measured in kilograms metres per second.
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This is the momentum at anytime ๐ก.
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But weโre only interested when ๐ก is three seconds.
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So we substitute ๐ก equals three here.
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๐ก is three and so 12๐ก is 36.
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And probably, the most sensible thing to do is just to put this into our calculator to get a momentum of 44550 kilogram metres per second.
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Thatโs the momentum of our car after three seconds.
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And as this car is moving in a straight line and the momentum turned out to be positive, itโs also the magnitude of the carโs momentum.
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This is therefore the answer to our question, 44550 kilogram metres per second.