P'(x) = 0 implies that the rate of change of speed in the y direction is constant, not that the rate of change of speed in the x direction is constant, so this isn't correct.

Condition 1 requires that P(0) = 0 and P(L) = h. The diagram appears to also require that P'(0) = 0 and P'(L) = 0. So we have:

P(0) = d = 0

P'(0) = c = 0

So P(x) = ax^3 + bx^2

P(L) = aL^3 + bL^2 = h

P'(L) = 3aL^2 + 2bL = 0

From the P'(L) condition we get that b = -(3L/2)a

So:

P(L) = aL^3 - (3L/2)a*L^2 = h

or

a = -2h/L^3

b = -(3L/2)*(-2h/L^3) = 3h/L^2

Thus

P(x) = -(2h/L^3)x^3 + (3h/L^2)x^2

has the required properties.

-Dan