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DISCUSSION and HINTS
Shown below is a simulation of an experiment of dropping a ball onto a fixed surface. Before impact and after impact, energy is conserved. During impact, a large impact force is generated. Although we cannot calculate the details of the force during impact, we can use the impulse/momentum equation to calculate the average force during impact. Note that this experiment differs from your problem in terms of the duration of the impact, with your impact duration being only 20% of the duration shown here.
Parts (a)-(c)
Draw two FBDs for the duration of time that includes only the impact: one FBD with weight, and one without weight. Sum forces on the particle in the vertical direction. Use the linear impulse/momentum equation relating the change in linear momentum during the impact to the impulse of the forces acting on the particle during impact. Compare your with and without gravity answers.
Part (d)
Although energy is not conserved during impact, energy is conserved prior to and after impact. Use conservation of energy before and after impact to relate h1 and h2.