The hand needs to stop the egg from its motion. Thus it need to apply a force with is opposing the egg initial motion. This force is drawn on the figure below attached.
The expression of external applied force written in terms of time rate of momentum variation is called in physics the theorem of momentum. It is an equivalent way to write the second law of Newton.
`F_("hand") = (Delta(P))/(Delta(t)) = (P_f-P_i)/(t_f-t_i)` (0)
if the external applied force is constant (`F=constant` ) one can write the expression of instantaneous momentum as
`P(t) =F*t` (1) considering both P and F as vectors (time t is a scalar)
By applying finite differences ( `Delta` ) one gets
`P(t_2) -P(t_1) =F*(t_2-t_1)` or `P_2-P_1 =F*(t_2-t_1)`
which rearranged gives `F = (P_2 -P_1)/(t_2-t_1) =(Delta(P))/(Delta(t))` (2)
By differentiating (1) one obtains directly
`dP =F*dt` which gives `F =(dP)/(dt)` (3)
Since the force is the same (constant) from (2) and (3) we obtain
`(dP)/(dt) = (Delta(P))/(Delta(t))`
In only a few words, finite differences are equal to differentials for linear variable functions (P is a linear variable function of t for F constant).
From the expression (2) above (which is written for vectors P and F constant) by multiplying both terms of the equality with the time variation `Delta(t)` (which is a scalar) one obtains
`Delta(P) = F*Delta(t)`