Kinematic Equations for Uniformly Accelerated Motion

For the common and important special case of uniformly accelerated motion — motion in which the acceleration $a$ has one constant value throughout the interval considered — three standard relations connect initial velocity $u$, final velocity $v$, displacement $s$, acceleration $a$ and time $t$. As required by the WBCHSE syllabus, we derive all three both graphically (from the v-t graph of Section 2.7) and by calculus (by integrating the defining relations $a = dv/dt$ and $v = dx/dt$).

First equation: $v = u + at$

Graphical derivation. On a v-t graph, a body with constant acceleration $a$ has a straight-line graph of constant slope $a$, starting at $v = u$ when $t = 0$. The slope of a straight line equals (rise)/(run), so

Calculus derivation. Since $a$ is constant, $a = dv/dt$ can be directly integrated with respect to time from $0$ to $t$ (with velocity going from $u$ to $v$):

Second equation: $s = ut + \tfrac{1}{2}at^2$

Graphical derivation. From Section 2.7, displacement equals the area under the v-t graph — here a trapezium with parallel sides $u$ and $v = u+at$, and width $t$:

Calculus derivation. Since $v = dx/dt$ and, from the first equation, $v(t) = u + at$, we integrate position from $x_0$ (taken as the origin, $s = x - x_0$) as $t$ runs from $0$ to $t$:

This is also, as noted in Section 2.6, exactly why the position-time graph of uniformly accelerated motion is a parabola: $s$ is a quadratic function of $t$.

Third equation: $v^2 = u^2 + 2as$

Graphical derivation. Eliminate $t$ between the first two equations: from $v = u + at$ we get $t = (v-u)/a$; substituting into $s = ut + \tfrac{1}{2}at^2$ and simplifying algebraically gives $v^2 = u^2 + 2as$.

Calculus derivation (chain rule). Using $a = dv/dt$ and the chain rule $\dfrac{dv}{dt} = \dfrac{dv}{dx}\dfrac{dx}{dt} = v\dfrac{dv}{dx}$, we can write the constant acceleration as $a = v\,dv/dx$, i.e. $a\,dx = v\,dv$. Integrating both sides — position from $0$ to $s$, and velocity from $u$ to $v$:

(This chain-rule derivation is worked out in full generality in Numerical 4.) …