Can I apply Newton's equations of motion to relative motion?

Question

We know that
Velocity of A relative to B is
$$ vec v_{A|B} = vec v_A - vec v_B $$
and Acceleration of A relative to B is
$$ vec a_{A|B} = vec a_A - vec a_B $$
So, is it correct to do this to find the displacement of A relative to B?:-
$$ vec S_{A|B} = (vec u_A - vec u_B) t + 0.5 (vec a_A - vec a_B) t^2 $$

Dale · Accepted Answer

Yes $$vec S_A = vec u_A t + 0.5 vec a_A t^2$$ $$vec S_B = vec u_B t + 0.5 vec a_B t^2$$ so then $$vec S_{A|B}=vec S_A - vec S_B$$ recovers your expression.

Perelman92 · Answer

don't forget to specify three things:

the velocity you mentioned is initial velocity, so better to call them $v_{A0}$ and $v_{B0}$
this is true only for constant acceleration, otherwise we have to consider the function that describes it
A and B begin to accelerate at the same time respect to $t_0$.

Stated that, if acceleration is not constant but is a function of $t$ or even $s$ you have to integer them separately, and you lost the incisiveness of the formula you wrote

Can I apply Newton's equations of motion to relative motion?

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