We can solve this problem using the kinematic equations of motion for a projectile. We can assume that the marble has zero initial vertical velocity, since it is rolling off horizontally.

(a) To find the time it takes for the marble to reach the floor, we can use the kinematic equation:

y = viyt + 1/2ay*t^2

where y is the initial vertical displacement (1.5 m), viy is the initial vertical velocity (zero), ay is the acceleration due to gravity (-9.81 m/s^2), and t is the time. Solving for t, we get:

t = sqrt(2y/ay) = sqrt(2(1.5 m)/9.81 m/s^2) ≈ 0.55 s

So it takes about 0.55 seconds for the marble to reach the floor.

(b) To find the initial horizontal velocity (vx) and initial vertical velocity (vy), we can use the kinematic equation:

x = vixt + 1/2ax*t^2

where x is the horizontal displacement (2.0 m), vix is the initial horizontal velocity, and ax is the acceleration in the x direction (zero, since there is no horizontal acceleration). Solving for vix, we get:

vix = x/t = 2.0 m/0.55 s ≈ 3.64 m/s

So the initial horizontal velocity of the marble is approximately 3.64 m/s.

Since the initial vertical velocity (vy) is zero, we can use the kinematic equation:

y = viyt + 1/2ay*t^2

where y is the vertical displacement (1.5 m), viy is the initial vertical velocity (zero), and ay is the acceleration due to gravity (-9.81 m/s^2). Solving for viy, we get:

viy = (y - 1/2ayt^2)/t = (1.5 m - 1/2*(-9.81 m/s^2)*(0.55 s)^2)/0.55 s ≈ 8.44 m/s

So the initial vertical velocity of the marble is approximately 8.44 m/s (upwards).