WEBVTT
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A particle moves along the ๐ฅ-axis.
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The velocity of the particle at time ๐ก is three ๐ก squared plus five ๐ก.
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What is the total distance travelled by the particle from time ๐ก equals zero to time ๐ก equals four?
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Weโre told that the particle moves along the ๐ฅ-axis.
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This is a single straight line.
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So we can use the rules for rectilinear motion.
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Letโs begin by recalling the relationship between velocity and displacement when considering rectilinear motion.
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Now I said displacement as velocity is a vector quantity, as is displacement.
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Distance is a scalar quantity.
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So weโll soon use the fact that itโs the magnitude of the displacement.
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Velocity is the rate of change of displacement over time.
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In other words, if ๐ is a function for displacement in time, then ๐ฃ, velocity, is equal to d๐ by d๐ก.
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Itโs the derivative of the function ๐ for displacement with respect to time.
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We can reverse this idea and use the fact that integration is the reverse process for differentiation.
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We can say that ๐ is equal to the integral of the function ๐ฃ with respect to ๐ก.
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To find the displacement of the particle between zero and four seconds then, we integrate our expression for velocity with respect to time between the limits of zero and four.
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Remember, when we integrate, we add one to the power and then divide by that new number.
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So the integral of three ๐ก squared is three ๐ก cubed divided by three, which is just ๐ก cubed.
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And the integral of five ๐ก is five ๐ก squared divided by two.
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And of course, weโre going to evaluate this between zero and four.
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Thatโs four cubed plus five times four squared over two minus zero cubed plus five times zero squared over two.
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Zero cubed is zero, and five times zero squared divided by two is zero.
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So we get 64 plus five times 16 over two.
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We divide through by two here.
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And we see that the displacement becomes 64 plus 40, which is 104 or 104 units.
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Remember, weโre trying to find the distance travelled during this time.
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And distance is equal to the magnitude of the displacement.
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Well, the magnitude of 104 is just 104.
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So the total distance travelled by the particle from time ๐ก equals zero to time ๐ก equals four is 104 or 104 units.