To estimate the time it would take to fall 100 miles (approximately 160,934 meters), we can use the physics of free fall. Assuming the object falls freely under the influence of gravity (ignoring air resistance), we can use the following equation for distance \( d \) fallen under constant acceleration:
\[ d = \frac{1}{2} g t^2 \]
Where:
- \( d \) is the distance fallen (in meters),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)),
- \( t \) is the time in seconds.
Rearranging this to solve for time \( t \):
\[ t = \sqrt{\frac{2d}{g}} \]
Substituting \( d = 160,934 , \text{meters} \) and \( g \approx 9.81 , \text{m/s}^2 \):
\[ t = \sqrt{\frac{2 \times 160,934}{9.81}} \approx \sqrt{32,757.05} \approx 181.07 , \text{seconds} \]
Thus, under ideal conditions with no air resistance, it would take approximately 181 seconds to fall 100 miles.
However, in reality, air resistance significantly affects falling objects, particularly at high altitudes. Objects in free fall reach a terminal velocity, which would greatly increase the time taken to fall 100 miles. For a skydiver, terminal velocity is around 53 m/s (around 120 mph), which means the time to fall would be much longer in practical scenarios.
Thus, taking air resistance into account, the time to fall 100 miles would be significantly longer than the ideal calculation, likely on the order of several minutes.
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