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The question:

Two bodies move in a straight line towards each other at initial velocities $v_1$ and $v_2$ and with constant accelerations $a_1$ and $a_2$ directed against the corresponding velocities at the initial instant.
What must be the maximum initial separation $l_{max}$ between the bodies for which they meet during the motion?

This is the solution given in the book:

We shall consider relative motion of the bodies from the viewpoint of the first body. Then at the initial moment, the first body is at rest(it can be at rest at the subsequent instants as well), while the second body moves towards it at velocity $v_1+v_2$. Its acceleration is constant, equal in magnitude to $a_1+a_2$, and is directed against the initial velocity. The condition that the bodies meet indicates that the distance over which the velocity of the second body vanishes must be longer than the separation between the bodies at the beginning of motion; hence we obtain:
$$l_{max} = \frac{(v_1+v_2)^2}{2(a_1+a_2)}$$

According to me, the answer can be obtained without use of relative motion. The maximum separation should be equal to the sum of the maximum displacement of each body.
i.e., $l_{max} = \frac{v_1^2}{2a_1} + \frac{v_2^2}{2a_2}$

Why are the answers in both the methods different? Is there something wrong with my method?

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Your method is wrong because it assumes that both bodies will have zero absolute speed at the same moment. This is wrong generally and only happens if $v_1/a_1=v_2/a_2$. For this case your answer would coincide with the book's.

In all the other cases you should consider relative motion.

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