Building blocks

## Warm-up solution

You are moving along a straight line, with your acceleration given by $a(t) = 6t - 2.$

• How fast will you be travelling at $t=2$?

You can think about the relationship between acceleration, velocity and displacement in two ways.

$a(t) = \dfrac{d}{dt}\big(v(t)\big)$

$v(t) = \dfrac{d}{dt}\big(s(t)\big)$

$s(t) = \int{v(t)}\, dt$

$v(t) = \int{a(t)}\,dt$

So we need to integrate the acceleration function to find your velocity function.

\begin{align*} v(t) &= \int{(6t - 2)}\,dt\\ v(t) &= 3t^2 - 2t + c \end{align*}

What does the constant $c$ represent in terms of velocity?

We know that $t=2$. Do we have enough information to say how fast we are going?

If we calculate $v(2)$ we get

\begin{align*} v(2) &= 12 - 4 + c\\ &= 8+c, \end{align*}

so we could say our velocity is $8 + c$ units.

Or we could calculate the integral

\begin{align*} \int_{0}^{2}{(6t - 2)}\,dt &= \left[3t^2 - 2t +c \right]^2_0\\ &= v(2) - v(0) \\ &= 8 \end{align*}

and say our velocity is $8$ units faster than it was at $t = 0$.

In both cases we have accounted for the constant of integration which represents our initial velocity. This is important as we weren’t given any information about our velocity at $t=0$, so we cannot assume that it was zero.

• Where will you be on the line at $t=2$?

To then find your displacement function we need to integrate the velocity function.

\begin{align*} s(t) &= \int{(3t^2 - 2t + c)}\,dt\\ s(t) &= t^3 - t^2 + ct + d \end{align*}

Why is it important not to forget the constant of integration from the velocity function?

What does the constant $d$ represent in terms of displacement?

With the displacement function we could calculate

\begin{align*} s(2) &= 2^{3} - t^{2} + 2c + d \\ &=4 + 2c +d \end{align*}

or

\begin{align*} \int_{0}^{2}{(3t^2 - 2t + c)}\,dt &= \left[t^3 - t^2 + ct + d \right]^2_0 \\ &= s(2) - s(0) \\ &= 4 + 2c. \end{align*}

One of these gives us your displacement from your starting point, and one gives us your displacement relative to a fixed origin. Which is which?

Under what circumstances would the two answers be the same?