How Long Does a Ball Take to Drop from 750 Meters and the Role of Air Resistance

When a ball, regardless of its type or weight, is dropped from a height of 750 meters, the time it takes to hit the ground can vary significantly due to the effects of air resistance. Understanding the impact of these factors and the formulas behind them is crucial for accurately predicting the ball's descent.

Drop from Rest at 750 Meters

When dropped from rest, the time taken for the ball to fall from 750 meters to the ground is typically around 12.8 seconds, although this can vary depending on the type of ball. For instance:

A 42-pound cannon ball takes approximately 12.8 seconds. A 6-pounder ball takes around 13.2 seconds. A football, due to its shape and lower mass, takes about 38 seconds.

While these times are approximate, they provide a good indication of the impact of air resistance on the ball's descent.

Thrown Downward at 25 Meters per Second

When a ball is thrown down from a height of 750 meters with an initial velocity of 25 meters per second, the time taken to reach the ground decreases. In this case, the times vary as follows:

A 42-pound cannon ball takes approximately 10.6 seconds. A 6-pounder ball takes around 11 seconds. A football takes about 37 seconds.

The lighter and larger the ball, the more air resistance will affect its speed. Thus, the initial speed of the ball also influences the duration of the fall. To accurately calculate the descent time, air resistance must be accounted for, but for simplicity, the basic formula can be used to get a close estimate.

Explanation

If you ignore air resistance, the time taken for the ball to fall can be significantly underestimated. For a ball dropped from 750 meters, air resistance would dramatically slow it down, and without air resistance, the ball would reach a speed of approximately 122 meters per second (274 mph) by the time it hits the ground. This speed can be calculated using the equation:

mgh frac{1}{2}mv^2

Solving for v, we get:

v sqrt{2gh}

Substituting the values, we find:

v ≈ sqrt{2 times 10 times 750} ≈ 122 m/s

This speed is so high that the force of air resistance would be nearly a hundred times greater compared to being stuck out of a car window at 27 mph.

The time taken for a ball to fall in the presence of air resistance can be calculated using the equation:

t frac{v - u}{g}

For a ball dropped from rest, the initial velocity (u) is zero, so the equation simplifies to:

t frac{v}{g}

The height (h) can also be expressed as:

h ut frac{1}{2}gt^2

Given u0, we have:

2h gt^2

Solving for time (t), we get:

t sqrt{frac{2h}{g}}

The negative value for time is disregarded since time cannot be negative.