When the ball hits the inclined plane, its velocity has vertical and horizontal components. The coefficient of restitution (e = 3/4) applies only to the velocity component perpendicular to the plane. The velocity parallel to the plane remains unchanged.

First, find the vertical velocity just before impact: v_y = g * t = 9.8 * 2 = 19.6 m/s (downward).

Resolve this into components perpendicular (v_⊥ = v_y * cos30°) and parallel (v_∥ = v_y * sin30°) to the plane.

After impact, the perpendicular component becomes v'_⊥ = e * v_⊥ = (3/4) * v_⊥ (up the plane). The parallel component is unchanged.

The time to hit again depends on the motion perpendicular to the plane. The acceleration perpendicular to the plane is g * cos30° downward. The time of flight for this projectile motion is given by:

$T=\frac{2\times v_{\perp }\text{'}}{g\times \cos 30°}$

Substituting the values, T = 4 seconds.

Final Answer:

4 seconds