The kinetic energy the ball has when it hits the block is $mgh$. We have the relation:
$$
\frac{mv^2}{2}=mgh
$$
This can be rewritten:
$$
m^2v^2=2m^2gh
$$
Which means the downward momentum of the ball is:
$$
p=mv=m\sqrt{2gh}
$$
The block will excert a force on the ball to cancel this momemntum and give it a momentum of $e\cdot p$ in the opposite direction:
$$
F\Delta t=m\sqrt{2gh}(1+e)
$$
This will require a corresponding increase in the normal force on the block and thereby in the friction force:
$$
\Delta F_f=\mu\Delta N=\mu F
$$
The change in moementum of the block will be:
$$
\Delta p_B=\Delta F_f\Delta t=\mu F\Delta t=\mu m\sqrt{2gh}(1+e)
$$
The change (decrease) in speed of the block will thus be $\mu\sqrt{2gh}(1+e)$.
I have not included the extra force due to the weight of the ball or due to giving it a horizontal velocity component or spin, but these are likely to be small.
So here, really, lies my question: Is there even a point to arguing about this?
Perhaps there is a point in discussing this.
In the Newtonian point of view, impulse and change of momentum are different concepts. Why?
Force $F(t)$ is a basic quantity describing instantaneous influence of one body on another, in general having a magnitude and direction, but let's have everything in the same direction here for simplicity. The formula for force has to be inferred from other laws of physics - it can be due to gravity ($mg$), spring ($-kx$), or air resistance $(-cv^2)$ or others or their combination.
With this notion of force, impulse of the force $F$ in the time interval $t_1..t_2$ is defined as
$$
I = \int_{t_1}^{t_2} F(t) dt.
$$
One could calculate impulse without knowing anything about momentum.
Now, based on the 2nd law for body with constant mass (not definition here)
$$
F = m\frac{dv}{dt},
$$
we can derive that
$$
I = m \Delta v
$$
and since $m$ is constant, also
$$
I = \Delta (mv).
$$
which means that impulse equals change in momentum. This is the historic and common point of view, I believe.
Alternatively, if you take the point of view where "force" is defined as $ma$, then impulse and change of momentum of the body have the same values as a consequence of definitions only. But I wouldn't say this means that impulse and change of momentum are the same concepts, because they are introduced in a different way with different name and symbol. So either way, I think it is safe to say that both are different concepts, while having the same value, either approximately (if 2nd law is taken as approximate law of physics) or exactly (if it is taken as a definition of force).
Best Answer
Well, you are right about force due to gravity being ignored. But that's a good approximation. In case of explosion the velocity of fragments after explosion changes significantly in very small time. This significant change in momentum means high impulsive force. So impulse due to internal forces on fragments is very large in comparison to force due to gravitation. Similar is the case of an object hitting on the pan. Force due to change in momentum is very high compared to gravitational force, hence force due to gravity can be ignored in calculation. Try solving the problem you have, including gravitational force. You conclude yourselves if it is significant.