Final answer:
To fall from the top of the Empire State Building without an initial velocity, a marble would take approximately 8.8 seconds. To hit the ground in exactly 5 seconds, the marble would need an initial velocity of 22.95 m/s downward.
Step-by-step explanation:
The time it would take for a marble to fall from the top of the Empire State Building can be found using the free fall formula h = ½gt², where h is the height, g is the acceleration due to gravity (approximately 9.81 m/s²), and t is the time in seconds. By rearranging the formula, we can solve for t:
t = √(2h/g). Substituting the given values: t = √(2*381m/9.81 m/s²) = √(77.58 s²), which simplifies to approximately 8.8 seconds.
To make the marble hit the ground in 5 seconds, you would have to give it an initial velocity that shortens the fall time. Using the formula v = gt (for the time it takes to fall without an initial velocity), and noting that the marble needs to cover the same distance in a shorter time, we can rearrange the original free fall formula to include initial velocity v_i: h = v_it + ½gt². Solving for v_i gives v_i = (h - ½gt²)/t. Substituting in the values: v_i = (381m - ½*9.81 m/s²*(5 s)²)/(5 s), we find that the initial velocity required is 22.95 m/s downward.