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2. Since both players are the same height, the 'vertical displacement' defined is the same as the diff. in height between their shoe soles.
If 0 < t < t_R, Boris is stationary, and D(t) is simply the height of Alice's jump as a function of time i.e. D(t) = h_A(t) - 0 = ut - 0.5gt^2 .
(Note that acceleration here is -g, since we are measuring distance upwards as +ve, and gravity acts in the opposite direction).
[For a harder version of the problem, try to find D(t) when t > t_R.
Then,
h_A(t) = ut - (0.5)gt^2
h_B(t) = u(t - t_R) - (0.5)(g)(t-t_R)^2
= ut - ut_R - (0.5)(g)(t^2 - 2tt_R + t_R^2)
so D(t) = h_A(t) - h_B(t)
= ut - (0.5)gt^2 - (ut - ut_R - (0.5)(g)(t^2 - 2tt_R + t_R^2))
= ut - (0.5)gt^2 - ut + ut_R + (0.5)(g)(t^2 - 2tt_R + t_R^2)
= ut_R + (0.5)(g)( - 2tt_R + t_R^2)) ]
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