# 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?

Nov 8, 2017

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

#### Explanation:

We have:

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

We know that:

$v = \frac{\mathrm{ds}}{\mathrm{dt}}$

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

$\frac{\mathrm{ds}}{\mathrm{dt}} = 3 {t}^{2} - 22 t + 24$

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

$\int \setminus \mathrm{ds} = \int \setminus 3 {t}^{2} - 22 t + 24 \setminus \mathrm{dt}$

Which we can directly integrate, to get:

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

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

$0 = 0 - 0 + 0 + C \implies C = 0$

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

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