Calculating the Height of a Building Using Free Fall Principles

When a stone is dropped from the top of a building, it accelerates due to gravity until it reaches the ground. This common scenario can be analyzed using the principles of free fall and kinematics. This article explains how to calculate the height of a building based on the stone's speed when it hits the ground, using the SUVAT equations. We'll explore the calculations step-by-step.

Understanding the Problem

The problem states that a stone is dropped from the top of a building and hits the ground traveling at 45 meters per second. We need to determine the height of the building. We assume the stone falls from rest, and the acceleration due to gravity is 9.8 m/s2.

Using Kinematic Equations for Free Fall

We start by applying the kinematic equation for free fall, which describes the final velocity V of an object subjected to constant acceleration:

v u g*t

Given:

u 0 (the stone is dropped from rest) v 45 m/s (the stone's speed when it hits the ground) g 9.81 m/s2 (the acceleration due to gravity)

First, let's solve for the fall time t:

v u g*t

45 0 9.81t

t 45 / 9.81 seconds ≈ 4.587 seconds

Calculating the Height of the Building

The height h of the building can be found using the equation for the distance traveled under constant acceleration:

s ut 1/2 g t2

Since u 0, the equation simplifies to:

s 1/2 g t2

Now, substituting the known values:

s 1/2 * 9.81 * (4.587)2

s ≈ 1/2 * 9.81 * 21.0256 ≈ 103.2075 meters

Rounded to three significant digits, the height of the building is approximately 103 meters.

Exploring Additional Scenarios

Understanding the principles of free fall and kinematics can help solve various problems. For instance, consider the scenario where a stone's speed at a certain time is 49 m/s:

Vt V0 g*t

Given:

V0 0 (since the stone is dropped from rest) g 9.8 m/s2 Vt 49 m/s

Solving for time t:

49 0 9.81t

t 49 / 9.81 ≈ 5 seconds

The distance traveled in this case is:

St V0*t 1/2 g*t2 0 1/2 * 9.81 * 52 122.5 meters

Another example involves a stone taken for a time t of 3.85 seconds. Using the average velocity, the height of the building can be calculated:

v g*t 9.81 * 3.85 ≈ 37.74 m/s

Average velocity is:

v g*t/2 37.74/2 ≈ 18.87 m/s

The height of the building is:

h v*t 18.87 * 3.85 ≈ 72.66 meters

Conclusion

By understanding the principles of kinematics and applying the SUVAT equations, we can solve real-world problems involving free fall. These principles are crucial for various fields, including physics and engineering. For more resources and explanations, please visit my Quora Profile.