Time for a 12 Oz Lead Ball to Reach the Mariana Trench – An Exact Calculation

Every once in a while, a question surfaces that combines the vastness of the ocean with the physics of objects in motion. How long would it take for a 12 oz lead ball to reach the bottom of the Mariana Trench if dropped from the surface during the month of March? This article aims to dissect and provide exact calculations for such a scenario, using March as a reference point for convenience.

Depth of the Mariana Trench

The Mariana Trench is one of the deepest parts of the ocean, located in the western Pacific Ocean. At its deepest point, known as the Challenger Deep, it reaches a depth of approximately 36,000 feet or 10,973 meters. This extreme depth poses an interesting challenge when considering the free-fall motion of an object.

Acceleration Due to Gravity

When an object is dropped into a gravitational field, such as the earth, it accelerates due to gravity. On Earth, this acceleration is approximately 9.81 m/s2. However, this acceleration is not constant throughout the descent since the object eventually reaches its terminal velocity, which is discussed in the next section.

Terminal Velocity

Terminal velocity is the constant speed that an object reaches when the force of gravity is balanced by the resistance of the fluid (in this case, water). For a 12 oz lead ball falling through water, the terminal velocity can range from 1.5 m/s to 2 m/s. For our calculation, we will use an average terminal velocity of 1.5 m/s.

Time Calculation

To calculate the time it would take for the lead ball to reach the bottom of the Mariana Trench, we can use the terminal velocity. We will follow these steps:

Convert the depth from feet to meters: Calculate the time using the formula: Time Distance / Velocity. Converting the depth to meters: 36,000 feet ≈ 10,973 meters Using an average terminal velocity of 1.5 m/s: Time 10,973 m / 1.5 m/s ≈ 7,315 seconds Converted to hours: ≈ 2.03 hours

Therefore, it would take approximately 2 hours and 3 minutes for a 12 oz lead ball to reach the bottom of the Mariana Trench if dropped from the surface, assuming it quickly reaches terminal velocity and maintains that speed throughout the descent.

Neglecting Acceleration

In our initial calculation, we assumed that the lead ball quickly reaches terminal velocity and maintains it throughout the journey. However, in reality, the ball would experience an initial acceleration phase as it falls through the water. The depth, water density, and temperature all play a role in determining the exact time and terminal velocity.

Impact of Water Properties

The terminal velocity of a falling object through water is dependent on various factors, including the water density, temperature, and pressure. Using the formula Vt sqrt(2mg / CdρA), where:

m 12 oz 0.34 kg g 9.81 m/s2 Cd 0.47 (drag coefficient for a spherical object) ρ density of seawater 1030 kg/m3 A surface projection of the ball πr2

By solving for r from the volume of the ball:

ρpb 10,000 kg/m3 A 0.00127 m2

This results in a terminal velocity (Vt) of approximately 3.3 m/s. Therefore, the journey to the bottom at around 11,000 meters depth would take 3,333 seconds or 55′30″.

In conclusion, while the exact time varies based on the assumptions made, the calculation provides a realistic timeline. The Mariana Trench remains a fascinating area of study, offering endless opportunities for scientific exploration and curiosity.