If you were to jump off the Empire State Building how long would it take to hit the ground?
Answer: Difficult to calculate, but under 8.81 seconds.
Calculating how long it takes an object to fall a certain distance is a difficult question if you want an exact answer, but if you're willing to look at a simplified model (ignoring factors such as air resistance/frictions, which would be considerable), a crude approximation can be reached.
The acceleration due to gravity of an object is 9.81 m/s2 (meters per second per second) down. The amount of displacement resulting from a given acceleration over time is .5at2, where t is the amount of time (in seconds). The amount of displacement we need to equal is the height of the roof of the Empire State Building, 381 m (1250 ft).
After substituting our values into the displacement/acceleration equation, giving us -381 = .5(-9.81)t2. We can find our amount of time by isolating t. Doing this give us our final result of 8.81 seconds. If we assume that non-gravity factors would not be accelerating your decent, we can treat this as an upper limit to the result in real life. if you try you cant go past the bars
How ever if you decide to use feet units you come up with 6.154 seconds. Acceleration due to gravity is approximately -32.174 feet per second2. The Empire State building is 1250 feet tall. So the height, Y, at time, t, 0 is 1250.
The acceleration equation of the scenario would at first Y"=-32.174t. When we integrate this to find the velocity equation we recieve: Y'=-16.087t2+C. The integrating constant, C, would represent a starting velocity. Since the person "fell" off the building the initial velocity is 0, so Y'=-16.087t2. In order to find the position equation we integrate once more, yielding: Y=-5.36233t3+C. The integrating constant, C, in this case represents initial position which we know is 1250. So the position equation is: Y=-5.36233t3+1250. To find out when he hits the ground we solve for when height, Y, equals 0. 0=-5.36233t3+1250. This yields a time value of 6.1544 seconds.
*Note: This answer also ignores air resistance.
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