Problem 11 - An Introduction to Mechanics

Question

It is told that during World War II the Russians, lacking sufficient parachutes for airborne operations, occasionally dropped soldiers inside bales of hay onto snow. The human body can survive an average pressure on impact of \(30 \mathrm{lb} / \mathrm{in}^{2}\). Suppose that the lead plane drops a dummy bale equal in weight to a loaded one from an altitude of \(100 \mathrm{ft}\), and that the pilot observes that it sinks about \(2 \mathrm{ft}\) into the snow. If the weight of an average soldier is \(180 \mathrm{lb}\) and his effective area is \(5 \mathrm{ft}^{2}\), is it safe to drop the men?

Step-by-Step Solution

Answer

It is safe to drop the men as the impact pressure (0.25 lb/in²) is below the tolerance level (30 lb/in²).

Step 1: Calculate Average Pressure on Impact

Given the sink distance into the snow is 2 ft, and initial drop height is 100 ft, the total distance for energy dissipation is 2 ft. Since pressure is force per unit area, we calculate the force first. The weight of the soldier is 180 lb, thus the force associated with the impact is 180 lb. The pressure exerted can be calculated using the formula: \( P = \frac{F}{A} \), where \( P \) is pressure, \( F \) is force (weight of the soldier), and \( A \) is the area of impact. With \( F = 180 \mathrm{lb} \) and \( A = 5 \mathrm{ft}^{2} \), we get: \[ P = \frac{180}{5} = 36 \mathrm{lb/ft}^2. \] To convert \( \mathrm{lb/ft}^2 \) to \( \mathrm{lb/in}^2 \), we use the fact that \( 1 \mathrm{ft}^2 = 144 \mathrm{in}^2 \), therefore: \[ P = \frac{36}{144} \mathrm{lb/in}^2 = 0.25 \mathrm{lb/in}^2. \]

Step 2: Evaluate Safety Level of Pressure with Tolerance

The tolerable pressure for the human body is given as 30 \(\mathrm{lb/in}^2\). Since the calculated pressure on impact is \( 0.25 \mathrm{lb/in}^2 \), this is below the maximum acceptable pressure for the human body. Thus, the pressure experienced by a soldier upon impact in these conditions is well below the safe threshold.

Key Concepts

Pressure Calculations

When it comes to understanding how pressure impacts the body, we start by calculating the force involved. This is crucial because pressure is defined as the force exerted per unit area. In our exercise, the force is the soldier's weight, which is noted as 180 pounds. We then consider the area over which this force is distributed, given as 5 square feet. This gives us the initial pressure calculation formula:

• Pressure Formula: \( P = \frac{F}{A} \)
• Where: \( P \) = Pressure, \( F \) = Force (weight of the soldier), \( A \) = Area

Substituting the values, we get \( P = \frac{180}{5} = 36 \mathrm{lb/ft}^2 \).
Since pressure is often measured in pounds per square inch (psi), we convert it using: \( 1 \mathrm{ft}^2 = 144 \mathrm{in}^2 \). This gives us \( P = \frac{36}{144} \mathrm{lb/in}^2 = 0.25 \mathrm{lb/in}^2 \).
This step shows how differences in units can affect our interpretations of force and pressure.

Energy Dissipation

Next, let’s explore how energy dissipation factors into impact mechanics. When a soldier is dropped from a height, potential energy is transformed into kinetic energy. As the soldier hits the snow, this kinetic energy must be absorbed, largely by the snow, which represents the soft landing medium.

• The height from which they are dropped is 100 feet.
• As they descend, gravitational potential energy is released, calculated as \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is height.
• On impact, the kinetic energy is managed over the 2 feet sink distance into the snow.

The snow’s ability to absorb energy softens the landing, reducing the potential forces on the soldier’s body as the energy spreads over a longer period and distance. This is why understanding how energy is dissipated is crucial in calculating the impact pressure and ensuring the soldier's safety.

Safety Analysis in Physics

Safety analysis begins with understanding the physical tolerances of the human body under pressure. In this scenario, the calculated impact pressure experienced by the soldier is \( 0.25 \mathrm{lb/in}^2 \), well below the human body's tolerable impact pressure of \( 30 \mathrm{lb/in}^2 \).

• Through physics, we confirm that this pressure is significantly safer, as only a small fraction of the maximum limit is exerted.
• This analysis ensures that, despite the seemingly dangerous method of descent, the forces involved are within safe human thresholds.
• This involves calculations not only of the pressure but also consideration of energy dissipation factors that affect the pressure at impact.

By thoroughly understanding each aspect, the physics behind this exercise not only provides clarity on the solution but reassures that the method is safely within physical limits. This highlights the importance of detailed calculations and evaluations in ensuring the safety of such operations.