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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?:-

$$ \vec S_{A|B} = (\vec u_A - \vec u_B) t + 0.5 (\vec a_A - \vec a_B) t^2 $$

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    $\begingroup$ Note that ALL motion is relative. When you define your velocity, you must measure that velocity with respect to some other object (e.g., a road surface, another vehicle, etc.). $\endgroup$
    – David White
    Commented Jan 30, 2021 at 16:57
  • $\begingroup$ both acceleration must be constant for the displacement formula to be applicable $\endgroup$
    – Alex P
    Commented Jan 31, 2021 at 9:42

2 Answers 2

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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.

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don't forget to specify three things:

  1. the velocity you mentioned is initial velocity, so better to call them $v_{A0}$ and $v_{B0}$
  2. this is true only for constant acceleration, otherwise we have to consider the function that describes it
  3. 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

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