The Law of Falling Bodies
Introduction
The Law of Falling Bodies, derived from the pioneering work of Galileo Galilei, describes how objects behave when they fall towards the Earth under the influence of gravity. This law laid the foundation for classical mechanics and has profound implications in physics and engineering.Historical Context
Galileo's experiments in the late 16th and early 17th centuries challenged the prevailing Aristotelian view that heavier objects fall faster than lighter ones. Through careful observation and experimentation, Galileo concluded that, in the absence of air resistance, all bodies fall at the same rate, regardless of their mass.Key Concepts
1. Acceleration Due to Gravity
- The acceleration due to gravity (denoted as g) is approximately 9.81 m/s² on the surface of the Earth. - This means that, in the absence of air resistance, an object in free fall will increase its velocity by about 9.81 m/s every second.2. The Equation of Motion
The motion of a falling body can be described by the following equation:$$ s = ut + \frac{1}{2} a t^2 $$
Where: - s is the distance fallen (in meters) - u is the initial velocity (in m/s, which is 0 for a freely falling object) - a is the acceleration (for free fall, a = g) - t is the time of fall (in seconds)
For a body starting from rest, the equation simplifies to: $$ s = \frac{1}{2} g t^2 $$
3. Example Calculation
Consider an object dropped from a height of 20 meters. To find out how long it takes to hit the ground, we can rearrange the formula:1. Set s to 20 meters, g to 9.81 m/s², and u to 0:
$$ 20 = \frac{1}{2} \cdot 9.81 \, t^2 $$
2. Solve for t: - Multiply both sides by 2: 40 = 9.81 t² - Divide both sides by 9.81: \( t^2 = \frac{40}{9.81} \approx 4.08 \) - Take the square root: \( t \approx 2.02 \, s \)
Thus, it takes approximately 2.02 seconds for the object to hit the ground.