Particles P and Q, at the same height, exhibit equal gravitational potential energy, leading to the conservation of mechanical energy. Equal potential energy results in equal kinetic energy, implying the same speed.
Consider two particles, P and Q, moving vertically. At the instant when they are at the same height, their gravitational potential energy is the same. Since their potential energy depends solely on their height and not on their velocity, at this instant, both particles have the same gravitational potential energy.
Now, according to the law of conservation of energy, the total mechanical energy of a system is conserved when only conservative forces (like gravity) are acting.
As the potential energy decreases for both P and Q, an equal amount of energy is converted to kinetic energy. Since the potential energy is the same for both particles at the instant they are at the same height, their kinetic energies are also equal.
Moreover, as kinetic energy is proportional to the square of the velocity, equal kinetic energies imply equal speeds. Thus, at the instant when P and Q are at the same height, they have the same speed. Furthermore, since they have the same speed and are at the same height, their motions are symmetric, leading to them moving in opposite directions.