Question #e3d6f

Jan 26, 2018

$a = {F}_{\text{net}} / m$

Explanation:

Basically equation of motion is solving Newton's 2nd Law.

Assuming

${F}_{\text{net}} = m a$
$a = {F}_{\text{net}} / m$
$a = \frac{\mathrm{dv}}{\mathrm{dt}} = \frac{d}{\mathrm{dt}} \left(\frac{\mathrm{dx}}{\mathrm{dt}}\right) = \frac{{d}^{2} x}{{d}^{2} t}$

Unless we know the nature of F, we just solve for

$\Rightarrow \frac{{d}^{2} x}{{d}^{2} t} = a$

Otherwise, to get the proper solution, you need to solve
$\frac{{d}^{2} x}{{d}^{2} t} = {F}_{\text{net}} / m$

Perform integration twice and applying proper initial conditions
$\Delta x = \int \int a \left(t '\right) \mathrm{dt} ' \mathrm{dt}$
$\Rightarrow x = {x}_{0} + \int \int a \left(t '\right) \mathrm{dt} ' \mathrm{dt}$

If start from rest, ${v}_{0} = 0 \mathmr{and} a \left(t\right) = c o n s \tan t = a$

$\Rightarrow x = {x}_{0} + {v}_{0} t + \frac{1}{2} a {t}^{2} = {x}_{0} + \frac{1}{2} a {t}^{2}$

Of course, the exact form will depend on the nature of force is being applied too. For instant, if F is caused by a spring pulling on the mass, then solution involves a sinusoidal displacement.