Question

Two particles, A and B, are moving in a straight line. Take this to be the x-axis. At t = 0, particle A is trailing particle B, initially at some distance x behind B. Particle B

is accelerating at a constant 2.00 m/s2 and particle A is constantly accelerating at 3.00 m/s2. Particle A was able to overtake B after B has moved 50.0 m. How long does it take for particle A to overtake B?

Answer 1

Let's change the perspective and describes things in the non-inertial coordinate frame linked with particle A. In this coordinate frame particle $A$ has zero speed (and zero acceleration as well). Particle $B$ has the following acceleration:

$$a = 2m/s^2-3m/s^2 = -1m/s^2$$

This means, that particle $B$ moves back on the x-axis, toward particle $A$. Since it has moved $d = 50m$ with the constant acceleration, then the required time can be found from the kinematic equation of motion:

$$d = \dfrac{|a|t^2}{2}\\ t = \sqrt{\dfrac{2d}{|a|}} = \sqrt{\dfrac{2\cdot 50}{|-1|}} = 10s$$

Answer. 10s.