# Question ef1f0

Nov 12, 2017

$v \left(t\right) = \frac{2 t + 3}{\sqrt{2 {t}^{2} + 6 t}}$

If given values of $x$:

$v \left(t\right) = \frac{2 t + 3}{x}$ is simpler to use.

#### Explanation:

The key concept to understand here is that velocity represents a change in displacement. Mathematically, this is:

$v \left(t\right) = \frac{x \left({t}_{f}\right) - x \left({t}_{i}\right)}{\Delta t}$

If you know calculus (which you actually need to solve this problem) velocity is represented as a time derivative of displacement :

$v \left(t\right) = \frac{\mathrm{dx}}{\mathrm{dt}}$

You have been given a function that models displacement (x) at any given time (t). To get a velocity function, all you need to do is take a derivative with respect to time:

${x}^{2} = 2 {t}^{2} + 6 t$

This is a very basic power rule, with a small but of implicit differentiation:

$\frac{d}{\mathrm{dt}} \left({x}^{2}\right) = \frac{d}{\mathrm{dt}} \left(2 {t}^{2} + 6 t\right)$

$\implies 2 x \frac{\mathrm{dx}}{\mathrm{dt}} = 4 t + 6$

$\frac{\mathrm{dx}}{\mathrm{dt}} = \frac{4 t + 6}{2 x} = \frac{2 t + 3}{x}$

$\textcolor{b l u e}{v \left(t\right) = \frac{2 t + 3}{x}}$

Note that due to the nature of your function, your velocity is actually dependent on displacement (x) as well as time. If you don't want this, you can simply solve the original equation for $x$ and plug in:

${x}^{2} = 2 {t}^{2} + 6 t$

$\implies x = \sqrt{2 {t}^{2} + 6 t}$

color(blue)(v(t) = (2t + 3)/(sqrt(2t^2 + 6t))#

Depending on what information you're given, you can pick which of the simplifications you want to use.

I know I went through the actual calculus pretty quickly, so if you want some extra information on how to do that, check out these videos:

Hope that helped :)