Analytical solution:

We know Newton's Second Law is often written as:

F = ma .

However, that is not good enough here. What is the mass, m?  The mass of the empty cart?  The cart plus sand?  If so, at what time?

Instead, we use his law as Newton chose to express it originally:

F = dp/dt ,

where F is the force at some time t (a constant in this problem) and p is the total momentum of the system feeling the force, also evaluated at that time t. (Note that we can verify that

F = ma from this second version of Newton's law: If p = mv, and dp/dt =m*dv/dt, and we know that dv/dt = a, then we have F = ma again. Note that this argument relied on m being a constant though!)

Here is a slick way to use F = dp/dt to "solve the problem in one or two lines". We take the integral of both sides, we see that

∫ Fdt =P_f - P_i

The change in momentum is equal to the time integral of force. This the equation for "impulse". In this case, since force is just a constant, impulse is simply force multiplied by the time over which it acts:

impulse = ∫ Fdt = Ft

At the start of the problem, the initial momentum is zero:

Pi = 0

After a mass m has fallen, the final momentum is:

P_f = (M + m)v

Using m = bt, we can either construct our answer in terms of the fallen mass, m (as K&K wants) or in terms of the time, t.

Ft = Fm/b = P_f - P_i = (M + m) v

Solving for v, we get

v = F(m/b)/(M+m) = Ft / (M + bt)