Final answer:
The problem involves calculating the time a marble is in the air, its horizontal speed as it leaves a table, and its speed upon hitting the floor, all of which are determined using kinematic equations that consider acceleration due to gravity.
Step-by-step explanation:
The question is addressing a problem in projectile motion where a marble rolls off a tabletop, and we are asked to determine different aspects of its motion. The calculations involve the principles of kinematics under the influence of gravity, assuming no air resistance.
Part (a): How long is the marble in the air?
Using the formula t = √(2h/g), where h is the height (1.0 m) and g is the acceleration due to gravity (9.81 m/s2), the time the marble is in the air can be calculated.
Part (b): What is the speed of the marble when it leaves the table's edge?
The horizontal speed can be found using v = d/t, where d is the distance (3.0 m) and t is the time calculated in part (a).
Part (c): What is its speed when it hits the floor?
The speed when the marble hits the floor can be calculated by combining the horizontal speed and the final vertical speed, found using v = gt, via Pythagorean theorem.
From the given options, the correct one matches the calculation results: (c) 0.45 s, 4.43 m/s, 4.9 m/s.