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A particle started moving in a straight line from the origin such that its acceleration at time 𝑡 seconds is given by 𝑎 equals six 𝑡 minus two meters per second squared, where 𝑡 is greater than or equal to zero.
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Given that its initial velocity was 14 meters per second, determine its velocity 𝑣 and its displacement 𝑠 when 𝑡 equals two seconds.
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The velocity of the particle can be calculated by integrating the acceleration with respect to 𝑡.
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In a similar way, the displacement can be calculated by integrating the velocity with respect to 𝑡.
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In our example, to work out the velocity, we need to integrate six 𝑡 minus two.
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The integral of six 𝑡 is three 𝑡 squared.
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And the integral of two is two 𝑡.
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Therefore, 𝑣 is equal to three 𝑡 squared minus two 𝑡 plus 𝑐.
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In order to work out the value of the constant 𝑐, we need to substitute in the initial conditions.
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When 𝑡 equals zero, the velocity 𝑣 was equal to 14.
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Substituting in these values gives us a value of 𝑐 equal to 14.
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This means we can calculate the velocity at any time using the equation three 𝑡 squared minus two 𝑡 plus 14.
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In our example, we are asked to work out the velocity when 𝑡 equals two.
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So we need to substitute 𝑡 equals two into the equation.
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This gives us 𝑣 is equal to three multiplied by two squared minus two multiplied by two plus 14.
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Three multiplied by two squared is equal to 12.
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And negative two multiplied by two is negative four.
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Therefore, 𝑣 is equal to 12 minus four plus 14.
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This gives us a value for 𝑣, when 𝑡 equals two, of 22.
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The velocity when 𝑡 equals two seconds is 22 meters per second.
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The second part of the question asked us to work out the displacement when 𝑡 equals two seconds.
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In order to do this, we need to integrate three 𝑡 squared minus two 𝑡 plus 14.
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The integral of three 𝑡 squared is 𝑡 cubed.
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The integral of two 𝑡 is 𝑡 squared.
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And the integral of 14 is equal to 14𝑡.
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Therefore, 𝑠 is equal to 𝑡 cubed minus 𝑡 squared plus 14𝑡 plus 𝑐.
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As the particle started moving from the origin, we know that when 𝑡 is equal to zero, 𝑠, the displacement, is also equal to zero.
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Substituting these values into the equation gives us a value of 𝑐 equal to zero.
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We can work out the displacement of the particle at any given time 𝑡 using the equation 𝑠 equals 𝑡 cubed minus 𝑡 squared plus 14𝑡.
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Once again, we need to substitute 𝑡 equals two into this equation.
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This gives us 𝑠 is equal to two cubed minus two squared plus 14 multiplied by two.
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Two cubed is equal to eight.
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And two squared is equal to four.
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Eight minus four plus 28 is equal to 32.
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This means that the displacement of the particle when 𝑡 equals two seconds is 32 meters.
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After two seconds, the particle is 32 meters from the origin and is travelling with a velocity of 22 meters per second.