In a gravity-free hall, an experimenter of mass 50 kg is resting on...

By the principle of conservation of momentum, the initial momentum of the system is zero. When the experimenter pushes the pillow downwards, the experimenter moves upwards.

m_{pillow} v_{pillow} = m_\exp v_\exp

5 \times 8 = 50 \times v_\exp \Rightarrow v_\exp = 0.8 \text{ ft/s}

The time taken by the pillow to reach the floor is:

$t_1 = \dfrac{8 \text{ ft}}{8 \text{ ft/s}} = 1 \text{ s}$

During this time, the experimenter moves upwards. The height of the experimenter from the floor at $t_1 = 1 \text{ s}$ is:

H_\exp = 8 + v_\exp t_1 = 8 + 0.8 \times 1 = 8.8 \text{ ft}

The pillow undergoes a perfectly elastic collision with the floor, so it rebounds upwards with the same speed of $8 \text{ ft/s}$.

The relative velocity of the pillow with respect to the experimenter is:

v_{rel} = v_{pillow} - v_\exp = 8 - 0.8 = 7.2 \text{ ft/s}

The time taken for the pillow to reach the experimenter after rebounding is:

t_2 = \dfrac{H_\exp}{v_{rel}} = \dfrac{8.8}{7.2} = \dfrac{11}{9} \text{ s}

The total time elapsed in the process is:

$T = t_1 + t_2 = 1 + \dfrac{11}{9} = \dfrac{20}{9} \text{ s} \approx 2.22 \text{ s}$