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A particle started moving in a straight line.
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After 𝑡 seconds, its position relative to a fixed point is given by 𝑟 is equal to 𝑡 squared plus four 𝑡 minus one metres, where 𝑡 is greater than or equal to zero.
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Find the velocity of the particle when 𝑡 is equal to five seconds.
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Here, we’ve been given the position of a particle relative to a fixed point at a time 𝑡 seconds.
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This is a function of time.
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And this means the particle won’t necessarily travel at a constant velocity.
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We therefore need to find an expression for the velocity of the particle at a given time 𝑡 seconds.
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So let’s recall what we actually mean by the word velocity.
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It’s the change in the particle’s displacement over time.
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This means we can find a function for the velocity by differentiating the function for the displacement or the position of the particle relative to the fixed point with respect to time.
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So how do we differentiate the expression 𝑡 squared plus four 𝑡 minus one?
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We multiply each term by its power.
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And then we reduce that said power or exponent by one.
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So 𝑡 squared differentiates to two multiplied by 𝑡 to the power of one.
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That’s just two 𝑡.
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And four 𝑡 differentiates to one multiplied by four 𝑡 to the power of zero.
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And 𝑡 to the power of zero is one.
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So this differentiates to four.
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And the constant differentiates to zero.
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And this is because it’s currently negative one multiplied by 𝑡 to the power of zero.
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When we multiplied by that power of zero, we just get zero.
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So we can say that the velocity at 𝑡 seconds is given by the expression two 𝑡 plus four.
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And since the displacement was in metres and the time is in seconds, we say that the velocity is two 𝑡 plus four metres per second.
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To find the velocity when 𝑡 is equal to five seconds, we’ll substitute five into this expression.
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When 𝑡 is equal to five, 𝑣 is equal to two multiplied by five plus four.
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Two multiplied by five is 10.
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So the velocity is given by 14 metres per second.
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And at this stage, it’s useful to remind ourselves of a graphic that can help us remember how to relate displacement, velocity, and acceleration.
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We’ve already seen that we can differentiate an expression for displacement with respect to time to find an expression for the velocity.
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Similarly, we can differentiate an expression for the velocity with respect to time to form an expression for the acceleration.
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And since integration is the opposite of differentiation, we can find an expression for the velocity by integrating the expression for the acceleration with respect to time.
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And finally, we can integrate the expression for velocity with respect to time to find an expression for the displacement 𝑟.