Can we use relative motion if acceleration of two bodies are different?

Question

Suppose, on the $x$-axis, a body $A$ is moving with velocity $v_1$ and acceleration $a_1$, and a second body $B$ is moving with velocity $v_2$ and acceleration $a_2$. $B$ is at a distance of $S$ from $A$ ($v_1 \gt v_2$ and $a_2 \gt a_1$).

Can we use relative motion to find time here that when they will meet? [$A$ and $B$ are non-inertial frames, so how can we apply relative motion here?]

Answer 1

Yes. Set the position of particle B to be the origin of the non-inertial frame.

Initial position of A: $-S$

Initial velocity of A: $v_1 - v_2$.

And adding a Pseudoforce $- m_A a_2$ to A: $$ F'_A = m_A a_1 - m_A a_2\\ a'_1 = \frac{F'_A}{m_A} = a_1 - a_2. $$

The position of A:

$$ x' = -S + (v_1 - v_2) t + \frac{1}{2} (a_1 - a_2) t^2. $$

Answer 2

You apparently have the velocity and acceleration of objects (A) and (B) on an inertial frame. Apply the basic kinematic equations in that frame.