Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
This may be a stupid question, but in an assignment I was given recently I was given an equation for acceleration and told to use integration to work out the velocity after 8 seconds assuming that the device started from rest.
The equation for acceleration looked a little like this.
$$a = \frac{5t}{4} - 1$$
Is there a advantage to using integration over substituting into the formula $v=u+at$?
Like so.
\begin{align} v & = u + (\frac{5t}{4}-1)t \\\\ & = u + \frac{5t^2}{4} - t \end{align}
Yup, the main advantage is that integration gives you the right answer, while substituting in doesn't! More specifically, the formula $v = u + at$ applies in a constant acceleration problem, when $a$ doesn't depend on $t$. But here, $a$ does depend on $t$, so the formula doesn't apply.
The integration solution would look like: $$\begin{align} \frac{dv}{dt} &= \frac{5t}{4} - 1 \\ \int_0^8\frac{dv}{dt} \ dt &= \int_0^8\frac{5t}{4} - 1 \ dt\\ v(8) - v(0) &= \left[\frac{5t^2}{8} - t \right]_0^8\\ v(8) &= \frac{5\cdot 64}{8} - 8 \\ v(8) &= 32 \\ \end{align}$$
