Final answer:
To find how far the marble travels horizontally after falling off the table, calculate the time it is in the air using vertical motion equations, then use this time with the marble's initial horizontal velocity. The marble will travel approximately 3.08 meters.
Step-by-step explanation:
This involves calculating the time the marble is in the air, using principles of projectile motion and the equations of kinematics. First, we need to find the time the marble is in the air. Since the vertical motion is separate from the horizontal motion and is only affected by gravity, we start by using the kinematic equation for vertical displacement, s = ut + (1/2)at², where s is the vertical distance, u is the initial vertical speed (which is 0 m/s as the marble is rolled horizontally), a is the acceleration due to gravity (9.8 m/s²), and t is the time. Since the table is 1.3 m high, we have 1.3 = 0 + (1/2)(9.8)t². Solving for t, we find that the marble is in the air for approximately 0.514 seconds. Now that we know the time the marble is in the air, we can calculate the horizontal distance traveled using the horizontal speed of the marble. In the horizontal direction, we use the equation for distance, s = vt, where s is the distance, v is the horizontal velocity, and t is the time. With a horizontal speed of 6.0 m/s, the marble travels s = 6.0 m/s × 0.514 s ≈ 3.08 m. Hence, the marble will travel horizontally approximately 3.08 meters from the edge of the table to the floor.