The Science of Falling: Calculating Building Heights and Lunar Trajectories

Have you ever wondered how tall a building must be for someone to fall from it and take 1 minute and 22 seconds? Or, if a motorless rocket was launched, how long it would take to reach the Moon? These questions involve a blend of physics and mathematics, specifically the principles of gravitational acceleration and terminal velocity. Let's explore these fascinating topics and delve into the calculations behind them.

Calculating the Height of a Falling Person

In a hypothetical scenario, imagine a person falls off a building and takes 1 minute and 22 seconds (82 seconds) to reach the ground. If we assume the person maintains a constant falling speed during the entire fall, we can calculate the height of the building using the formula for distance traveled under constant acceleration:

Height 1/2 * g * t2

Where g is the gravitational acceleration (9.8 m/s2) and t is the time in seconds (82 seconds).

Height 1/2 * 9.8 m/s2 * (82 s)2

Height ≈ 33,896 meters

This height is significantly more than any known building on Earth, suggesting that the person must either have exceptionally high terminal velocity or the assumption that the fall speed is constant for the entire descent is incorrect.

The Role of Terminal Velocity

In reality, a human falling from a height would not maintain a constant speed. Instead, they would accelerate until reaching terminal velocity. Terminal velocity is the constant speed that a falling object reaches when the acceleration due to gravity is canceled out by the drag force from the air. For a person falling with both arms and legs outstretched, terminal velocity is approximately 54 meters per second (120 miles per hour).

To calculate the height of the building considering the terminal velocity, we first determine the time it takes to reach terminal velocity. Assuming uniform acceleration and taking the acceleration due to gravity as 9.8 m/s2, the time to reach terminal velocity (tterminal) can be calculated using the formula:

v u at

Where v is the terminal velocity (54 m/s), u is the initial velocity (0 m/s), and a is the acceleration due to gravity (9.8 m/s2).

54 0 9.8 * tterminal

tterminal ≈ 5.51 seconds

Next, we calculate the distance fallen during this accelerating period:

s 0.5 * g * tterminal2

s ≈ 0.5 * 9.8 * (5.51)2 ≈ 152.4 meters

After reaching terminal velocity, the person will continue at 54 meters per second for the remaining time. The remaining time (tremaining) is:

tremaining 82 - 5.51 ≈ 76.49 seconds

The distance fallen during this period (sremaining) is:

sremaining 54 * 76.49 ≈ 4111.46 meters

The total height of the building is the sum of these two distances:

Total height 152.4 4111.46 ≈ 4263.86 meters

This height is still significantly more than any known building on Earth, indicating that our initial assumption of a constant speed fall was incorrect.

Falling to the Moon: A Lunar Trajectory

For a less hypothetical scenario, consider a motorless rocket launched at NASA's Kennedy Space Center. How long does it take to reach the Moon? This involves understanding the escape velocity and trajectory of a rocket. The escape velocity from Earth's surface is approximately 11.2 km/s (or 11,200 m/s).

The time (t) to reach the Moon can be calculated using the formula for distance traveled under constant velocity:

Distance Velocity * Time

where the distance to the Moon is approximately 384,400 kilometers (384,400,000 meters) and the velocity is 11,200 m/s.

384,400,000 11,200 * t

t ≈ 34,385.71 seconds

This is approximately 9.55 hours. Note that this is a simplified calculation and doesn't account for gravitational pull from other celestial bodies or the need for a more complex orbital trajectory.

Conclusion

Falling from great heights and launching rockets to the Moon both involve complex physical principles, including gravitational acceleration and terminal velocity. By understanding these concepts, we can better appreciate the challenges and risks involved in both scenarios. The calculations we've explored here highlight the sheer scale of these events and the importance of precise physics in engineering and space exploration.