If the velocity of an object is given by $v(t) = 3t^2 - 22t + 24$ and $s(0)=0$ then how do you find the displacement at time $t$ ?

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Answer 1 · 2017-11-08T11:23:04

$s = t^3 - 11t^2 + 24t$

We have:

$v(t) = 3t^2 - 22t + 24$ and $s(0)=0$ ..... [A]

We know that:

$v = (ds)/dt$

So we can write [A] as a Differential Equation:

$(ds)/dt = 3t^2 - 22t + 24$

This is separable. "as is", so we can "separate the variables" to get:

$int \ ds = int \ 3t^2 - 22t + 24 \ dt$

Which we can directly integrate, to get:

$s = t^3 - 11t^2 + 24t + C$

Using the initial condition $s(0)=0$ we have:

$0 = 0 - 0 + 0 + C => C=0$

Giving uis a position function for the particle at time $t$ :

$s = t^3 - 11t^2 + 24t$

If the velocity of an object is given by v(t) = 3t^2 - 22t + 24 v(t) = 3t^2 - 22t + 24 and s(0)=0s(0)=0 then how do you find the displacement at time tt?