Calculating Distance Traveled in the Twelfth Second of a Free-Falling Object

Introduction

The question of how far a free-falling object moves during the twelfth second of its fall from rest might seem simple, but it requires careful consideration of various factors. The distance traveled by any object in free fall is influenced by the gravitational acceleration and the time it has been in motion. This article aims to provide a clear and accurate answer to this question, taking into account the specifics of gravitational acceleration and the impact of the time elapsed since the object began falling.

Gravitational Acceleration and Distance Calculation

When an object is in free fall, it accelerates due to the force of gravity. The standard value for gravitational acceleration on Earth is g 9.81 m/s2. The distance traveled by the object can be calculated using the formula for uniformly accelerated motion:

h frac{1}{2}gt^2

where h is the distance, g is the acceleration due to gravity, and t is the time.

Calculating the Distance for the 11th and 12th Seconds

To find out the distance traveled in the twelfth second, we need to calculate the distance traveled during the first 11 seconds and the first 12 seconds, and then find the difference between these two values.

Distance Traveled During the First 11 Seconds

The formula to calculate the distance traveled in the first 11 seconds is:

h_11 frac{1}{2}g(11)^2

Substituting g 9.81 m/s2, we get:

h_11 frac{1}{2} times 9.81 times (11)^2 604.095 text{ meters}

Distance Traveled During the First 12 Seconds

The formula for the distance traveled in the first 12 seconds is:

h_12 frac{1}{2}g(12)^2

Substituting g 9.81 m/s2, we get:

h_12 frac{1}{2} times 9.81 times (12)^2 704.88 text{ meters}

Distance Traveled in the 12th Second

The distance traveled specifically in the twelfth second can be found by subtracting the distance traveled in the first 11 seconds from the distance traveled in the first 12 seconds:

h_12 - h_11 704.88 - 604.095 100.785 text{ meters}

Alternative Approach Using Speeds

A more straightforward way to approach this problem is by considering the average speed during the twelfth second. The speed at the end of the 11th second is 11g, and the speed at the end of the 12th second is 12g. The average speed for the twelfth second is the average of these two speeds:

text{Average speed} frac{11g 12g}{2} 11.5g

Since the time is 1 second, the distance traveled in the twelfth second is:

text{Distance} 11.5g times 1 11.5 times 9.81 112.815 text{ meters}

Conclusion

The distance traveled during the twelfth second of a free-falling object can be accurately calculated using the provided formulas. Whether you use the distance formula or the average speed approach, the important factor is the understanding of gravitational acceleration and the time elapsed. This knowledge can be applied to various scenarios involving free-falling objects, from educational problems to real-world physics applications.