Final answer:
You'd need to solve a physics problem to determine the initial speed using kinematic equations, then convert speeds to the desired units. Time does run slightly faster at higher elevations due to general relativity, but not significantly enough to notice in every day life, thus not a reasonable excuse for being 5 minutes late.
Step-by-step explanation:
To determine how fast you need to throw a ball for it to reach the top of the Empire State Building, you would need to solve a physics problem involving projectile motion or simply use the kinematic equations if throwing it straight up. To calculate the speed, you can use the formula vf^2 = vi^2 + 2ad, where vf is the final velocity (which is 0 at the top of the building), vi is the initial velocity (what we are solving for), a is the acceleration due to gravity (-9.81 m/s^2, but since we are only considering the magnitude here, it's 9.81 m/s^2), and d is the distance to the top (1250 feet or about 381 meters). Solving for vi would give us the initial speed required. To convert the speed from feet per second to miles per hour, you would multiply by conversion factors: there are 0.3048 meters in a foot, 3,600 seconds in an hour, and 1 mile is 1,609 meters.
However, the question about the ball’s speed in feet per second and miles per hour to reach the top of the Empire State Building leads to an interesting consideration involving theoretical physics, specifically general relativity. According to general relativity, time does indeed run slightly faster at higher elevations. This is because a clock closer to a massive object (like the Earth) will experience a stronger gravitational field and therefore run slightly slower compared to a clock further away from the same massive object. So while your friend's assertion that time runs faster at the top of the Empire State Building is technically correct, the difference is minuscule and certainly not enough to account for a 5-minute discrepancy. Therefore, this excuse is not reasonable for being 5 minutes late.