Question

Derive the kinematic equation ∆x = vt + ½ at2 for constant acceleration. Explain all
steps and symbols used.

Answer 1

The constant acceleration $a$ results in the linear dependence of velocity $V$ on time $t$ . If $V_0$ is the initial velocity at the interval in question, therefore the dependence takes form

$V(t) = V_0 + at.$

Next let us draw a figure of $V(t)$.

The displacement is the area under curve representing the dependence of velocity on time. This is so because for every small interval of time $dt$ the displacement can be written as $V(t)\,dt$ (it's an area of a small rectangle) and if we add all the intervals together we'll get the area under $V(t)$ .

We can see that the area has a form of trapeze, so it's area can be calculated as

$\dfrac12(V_0 + (V_0+at))t = V_0t + \dfrac{at^2}{2}.$

So $\Delta x = V_0t + \dfrac{at^2}{2}.$