Calculating the Velocity of a Free-Falling Object: Understanding the Physics Behind Free Fall
Free-fall motion is a fundamental concept in physics, describing the motion of an object under the influence of gravity where air resistance is negligible. This article will guide you through the process of determining the velocity of a ball just before it hits the ground when it is dropped from a specific height.
Understanding Free-Fall Motion
In a vacuum, the acceleration of a ball dropped from a height is constant and equivalent to the acceleration due to gravity, denoted as g. For the purposes of this article, we assume the value of g to be 9.81 m/s2.
Calculating the Final Velocity
Given the formula for the final velocity of a free-falling object, we can calculate the velocity at which a ball hits the ground when dropped from a height of 380 meters. The relevant equation is:
v2 u2 2gh
Where:
v is the final velocity (in m/s) u is the initial velocity (in m/s) g is the acceleration due to gravity (in m/s2) h is the height from which the object is dropped (in m)Step-by-Step Calculation
1. **Identify the given values:**u 0 m/s (since the ball is dropped from rest) g 9.81 m/s2 h 380 m
2. Substitute these values into the equation:
v2 0 2 x 9.81 x 380
3. Simplify the equation:
v2 2 x 9.81 x 380 7449.6 m2/s2
4. Take the square root of both sides to find the final velocity:
v radic;7449.6 asymp; 86.3 m/s
Additional Examples and Solutions
Example 1: Dropping from 380 meters with a different equation
Another formula that can be used for the same calculation is:
v2 u2 2gh
Where:
v is the final velocity (in m/s) u 0 m/s (since the ball is dropped from rest) g 9.81 m/s2 h 380 mSubstitute the values into the equation:
v2 0 2 x 9.81 x 380 2 x 9.81 x 380 7449.6 m2/s2
Take the square root:
v radic;7449.6 asymp; 86.3 m/s
Example 2: Dropping from 300 meters
Using the same formula for a height of 300 meters:
v2 0 2 x 9.81 x 300 2 x 9.81 x 300 5880 m2/s2
Take the square root:
v radic;5880 asymp; 76.6 m/s
Conservation of Mechanical Energy
Another method to determine the velocity of a falling object is to use the conservation of mechanical energy. According to this principle, the total mechanical energy (kinetic potential) of a system is conserved. Hence, the potential energy at the height h is converted into kinetic energy just before hitting the ground.
frac12;m v2 mgh
Canceling m from both sides and taking the square root:
v radic;2gh radic;2 x 9.8 x 300 radic;5880 asymp; 76.7 m/s
Instantaneous vs. Gradual Velocity Change
The question arises whether the velocity of an object increases gradually or instantly during free fall. The answer is that the velocity changes instantly. The object gains velocity as soon as it is released due to the immediate action of gravity.
Note: The object's velocity will continue to increase by 9.81 m/s every second it is in free fall until it hits the ground.
In conclusion, the velocity of a ball dropped from a height of 380 meters is approximately 86.3 m/s, and from 300 meters, it is approximately 76.6 m/s. These calculations are based on the principles of free fall and the conservation of mechanical energy.