Question

Derive the equation s=ut+1/2at^2

Answer 1

We know

$a=\frac{dv}{dt}$

$dv=adt$

Integrate both sides

$\int\limits^v_u {} \, dv = \int\limits^t_0 {a} \, dt\\ v-u=at\\v=u+at$

We also know

$v=ds/dt\\ds=vdt$

Integrate both sides

$\int\limits^s_0 {} \, ds =\int\limits^t_0 {(u+at)} \, dt$

$s=\int\limits^t_0 {u} \, dt + \int\limits^t_0 {at} \, dt\\s=ut+ \frac{at^2}{2}$