Understanding the Time of Journey for a Stone in Free Fall

In this article, we will delve into the physics behind the journey of a stone falling freely from a height, specifically when it covers half the total distance in the last second of its descent. This is a common scenario in physics and can be analyzed using the equations of motion under uniform acceleration due to gravity. We will use these principles to derive the total time of the journey of the stone.

Equations of Motion and Free Fall

The equations of motion for a stone falling freely under gravity can be expressed using the following formula:

Distance fallen in t seconds:

Where g is the acceleration due to gravity, approximately 9.81 m/s2.

Falling Distance and Time Analysis

To determine the total time of the journey, t, we first need to analyze the distance traveled in the last second, which is given as half the total distance, h/2.

Total Distance Traveled in t Seconds

The total distance fallen from height h in t seconds is given by:

Distance Traveled in t-1 Seconds

The distance traveled in the t-1 seconds is:

Distance Traveled in the Last Second

The distance traveled in the last second from t-1 to t is:

Given the problem statement, this distance is h/2:

Rearranging the Equation

Rearranging the equation:

Solving for h:

Substituting into the solution:

Dividing by g (assuming g neq; 0).

Rearranging the equation:

Using the quadratic formula:

Calculating the solutions:

The valid solution for time (since time cannot be negative):

Therefore, the time of journey t for the stone falling from height h and traveling a distance h/2 in the last second is approximately 3.414 seconds.

Alternative Approach

Another approach to solve this problem:

By solving the equation, we find:

This confirms the total time of the journey.

Conclusion

The journey of a stone falling freely from a height, with the last second covering half the total distance, can be analyzed using the equations of motion. The derived time of journey provides insights into the physics of free fall and can be applied to various real-world scenarios.

Related Keywords

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