Question

what are the Newton equations of motion for constant acceleration.

Answer 1

We know that in an inertial reference frame the motion follows an equation

$ma = m\cdot \dfrac{d^2r}{dt^2} = F.$

The constant acceleration means $\dfrac{d^2r}{dt^2} =a$ , so

$v(t) = \dfrac{dr}{dt} = at+v_0$

and $r(t) = a\dfrac{t^2}{2} + v_0t + r_0.$

Here $v_0$ is the initial velocity and $r_0$ is the initial position.

The last equation can be rewritten in terms of the distance: $s(t) = a\dfrac{t^2}{2} + v_0t$ .

Here (https://thecuriousastronomer.wordpress.com/2014/11/11/newtons-equations-of-motion-revisited/) the author presents three additional equations ([3], [4], [5]), that can be obtained by rewriting the equation above in terms of the velocity after time t (v(t)). The most important of the three is [4], because it helps us to link the velocity and the distance without the time.