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I have a fairly simple question which I just can not figure out how to solve - this is the question:

A particle travels along a straight line with its velocity at time 't' seconds given by 'v' m/s where v = 4t + 1. Find the distance traveled in the fifth second.

Any help would be appreciated, thanks.

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    $\begingroup$ Integrate the expression and evaluate with with limits of 4 and 5 $\endgroup$ – David Quinn Jun 21 '15 at 10:58
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You should remember the following:

  • When we differentiate a function expressing the displacement of a particle, we get the function giving the velocity of the particle.

  • When we differentiate a function expressing the velocity of a particle, we get the function giving the acceleration of the particle.

As long as $t$ is present in the function we get upon Differentiating we can plug in $t=k$ to get the velocity or acceleration at time $t=k$.

  • Integration (with suitable limits) would perform the reverse operation and would express the displacement (when Integrating the Velocity Function) or the velocity (when Integrating the Acceleration Function) over a specific time period(ie the limits of the Interval of Integration).

$$\Rightarrow \text{Displacement over the time interval }t=4\text{ and }t=5 =\int_4^5 4t+1 dt$$ $$\bigg (2t^2+t\bigg )\bigg|^{t=5}_{t=4}$$

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