Understanding the Distance Traveled by a Freely Falling Body in One Second

This question may arise in a physics class, but how many times have we ever thought about the intricacies behind it, especially when paired with our AI team’s capabilities?

Calculating the Distance

The formula to calculate the distance (d) a freely falling body travels under the influence of gravity within a given time (t) is given by:

[ d frac{1}{2} g t^2 ]

where (g) is the acceleration due to gravity, approximately (9.81 , text{m/s}^2) near the Earth's surface.

For (t 1) second:

[ d frac{1}{2} times 9.81 , text{m/s}^2 times (1 , text{s})^2 ]

Carrying out the calculation:

[ d frac{1}{2} times 9.81 , text{m} approx 4.905 , text{m} ]

Therefore, a freely falling body will travel approximately 4.905 meters in one second.

Moving Deeper

In the first second of free fall, the body falls from an initial velocity (v 0 , text{m/s}) to (v 9.8 , text{m/s}).

The distance covered during this first second can be calculated by using the 'average velocity' concept:

[ text{Distance} text{Average Velocity} times text{Time} ]

Given that the uniform acceleration is constant at (9.8 , text{m/s}^2), the average velocity for the first second is:

[ text{Average Velocity} frac{0 , text{m/s} 9.8 , text{m/s}}{2} 4.9 , text{m/s} ]

Hence, the distance covered in the first second is:

[ text{Distance} 4.9 , text{m/s} times 1 , text{s} 4.9 , text{m} ]

Generalizing the Formula

For distance covered in any time interval, the formula can be generalized using the average velocity. The average velocity for any time interval (t) is:

[ v_{avg} frac{v_i v_f}{2} ]

At the start, (v_i 0 , text{m/s}), hence:

[ v_{avg} frac{0 gt}{2} frac{gt}{2} ]

The distance covered is then:

[ d v_{avg} times t frac{gt}{2} times t frac{1}{2}gt^2 ]

For intervals that do not start at (t 0), the initial velocity (v_i gt) where (t) is the start of the time interval of interest.

For example, between (t 3 , text{s}) and (t 4 , text{s}), the velocities are:

[ v_{3 , text{s}} 9.8 times 3 29.4 , text{m/s} ] [ v_{4 , text{s}} 9.8 times 4 39.2 , text{m/s} ]

The average velocity over this interval is:

[ text{Average Velocity} frac{29.4 , text{m/s} 39.2 , text{m/s}}{2} 34.3 , text{m/s} ]

Hence, the distance covered in this 1-second interval is:

[ text{Distance} 34.3 , text{m/s} times 1 , text{s} 34.3 , text{m} ]

Conclusion

The distance traveled by a freely falling body in one second is a fundamental concept in physics, applicable in various scenarios. Understanding this involves grasping the role of gravity and the importance of average velocity.

Whether this question arises in a physics class or a discussion on AI and engineering, it always provides valuable insights!