Analysis:
To find the velocity of the cart when the block drops on it, we need to consider the conservation of energy and momentum.

Conservation of Energy:
When the block slides down the inclined board, it gains potential energy due to its initial height h. As it reaches the bottom of the incline, all the potential energy is converted into kinetic energy.

The potential energy of the block at height h is given by:
Potential Energy = mgh

At the bottom of the incline, the potential energy is converted into kinetic energy:
Kinetic Energy = (1/2)mv^2

Since there is no friction, the kinetic energy gained by the block is equal to the kinetic energy of the cart when the block drops on it.

Conservation of Momentum:
When the block drops on the cart, the momentum of the system is conserved. The momentum of the block just before it drops is given by:
Momentum of the block = m * velocity of the block

The momentum of the cart just after the block drops on it is given by:
Momentum of the cart = M * velocity of the cart

Since momentum is conserved, we can equate the momentum of the block and the cart:
m * velocity of the block = M * velocity of the cart

Combining the Equations:
Equating the kinetic energy gained by the block and the kinetic energy of the cart, we have:
(1/2)mv^2 = (1/2)MV^2

Simplifying the equation, we get:
v^2 = (M/m) * V^2

Now, substituting the momentum equation into the equation above, we have:
v^2 = (M/m) * (m * velocity of the block)^2

Simplifying further, we get:
v^2 = M * (velocity of the block)^2

Taking the square root of both sides, we have:
v = √(M * (velocity of the block)^2)

Therefore, the velocity of the cart when the block drops on it is equal to the velocity of the block itself, multiplied by the square root of the ratio of masses M/m.

Conclusion:
The velocity of the cart when the block drops on it is determined by the initial velocity of the block sliding down the inclined board and the ratio of masses between the block and the cart. The conservation of energy and momentum principles allow us to derive this relationship.