Answer to Question #115192 in Mechanics | Relativity for ntule

Question #115192
Derive the kinematic equation ∆x = vt + ½ at2 for constant acceleration. Explain all
steps and symbols used.
Expert's answer

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}."

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