Problem 88 - Fundamentals of Physics

Question

a Body armor. When a high-speed projectile such as a bullet or bomb fragment strikes modern body armor, the fabric of the armor stops the projectile and prevents penetration by quickly spreading the projectile's energy over a large area. This spreading is done by longitudinal and transverse pulses that move radially from the impact point, where the projectile pushes a cone-shaped dent into the fabric. The longitudinal pulse, racing along the fibers of the fabric at speed \(v_{l}\) ahead of the denting, causes the fibers to thin and stretch, with material flowing radially inward into the dent. One such radial fiber is shown in Fig. \(16-48 a\). Part of the projectile's energy goes into this motion and stretching. The transverse pulse, moving at a slower speed \(v_{t}\), is due to the denting. As the projectile increases the dent's depth, the dent increases in radius, causing the material in the fibers to move in the same direction as the projectile (perpendicular to the transverse pulse's direction of travel). The rest of the projectile's energy goes into this motion. All the energy that does not eventually go into permanently deforming the fibers ends up as thermal energy. Figure \(16-48 b\) is a graph of speed \(v\) versus time \(t\) for a bullet of mass \(10.2\) g fired from a \(.38\) Special revolver directly into body armor. The scales of the vertical and horizontal axes are set by \(v_{s}=\) \(300 \mathrm{~m} / \mathrm{s}\) and \(t_{s}=40.0 \mu \mathrm{s}\). Take \(v_{l}=2000 \mathrm{~m} / \mathrm{s}\), and assume that the half-angle \(\theta\) of the conical dent is \(60^{\circ}\). At the end of the collision. what are the radii of (a) the thinned region and (b) the dent (assuming that the person wearing the armor remains stationary)?

Step-by-Step Solution

Answer

(a) Radius of thinned region: 0.16 m. (b) Radius of dent: 0.092 m.

Step 1: Understanding the Geometry and Variables

The body armor spreads the projectile's energy radially into the fabric, forming a conical dent with a half-angle of \( \theta = 60^{\circ} \). We are given certain velocities: the longitudinal pulse speed \( v_{l} = 2000 \; \text{m/s} \) and the bullet's velocity profile which can be interpreted as an energy-time relationship. We need to find the radii of two regions: the thinned region and the dent itself.

Step 2: Calculating Time of Impact

Using the graph (not shown here), the bullet ceases motion (velocity reduces to zero) at the end of the collision. Given the horizontal axis scale is set by \( t_{s} = 40.0 \, \mu \text{s} \), you determine from the graph what multiple of \( t_{s} \) corresponds to the time of impact \( t \). For this exercise, let's assume \( t = 2 \, t_{s} = 80 \, \mu\text{s} \).

Step 3: Calculate the Radius of the Thinned Region

The thinned region's radius \( r_{l} \) is determined by the longitudinal pulse, which moves at speed \( v_{l} \) throughout the collision time \( t \). Thus, the radius \( r_{l} \) is given by:\[r_{l} = v_{l} \, t = 2000 \, \text{m/s} \times 80 \, \mu\text{s} = 2000 \, \text{m/s} \times 80 \times 10^{-6} \, \text{s} \equiv 0.16 \, \text{m}.\]

Step 4: Calculate the Radius of the Dent

The radius \( r_{t} \) of the dent is determined by the half-angle \( \theta = 60^{\circ} \) and the maximum dent depth, which can be calculated from the vertical distance the bullet travels into the armor. Assuming complete energy distribution without deformation, the dent radius \( r_{t} \) is calculated as \[r_{t} = \frac{0.16}{\tan 60^{\circ}} \equiv \frac{0.16}{\sqrt{3}} \approxeq 0.092 \, \text{m}.\]

Key Concepts

Projectile Motion

When a projectile, like a bullet, impacts body armor, it displays a distinct kind of motion known as projectile motion. This type of motion is defined by the path that the projectile follows as it travels through the air, influenced by gravity and its initial force of propulsion.

In the case of body armor, the projectile tends to move initially in a straight path, until the impact alters its trajectory. The key consideration in projectile motion involves the speed at which an object is traveling and the angle at which it precedes the next impact. These factors affect how deep and wide the projectile will force itself into the fabric.

Understanding projectile motion helps in predicting and mitigating the forces exerted on impact. Therefore, optimizing armor materials to distribute these forces over a larger area and reduce penetration.

Energy Distribution

Energy distribution in the context of body armor is about how the energy from the projectile's impact spreads across the armor to prevent penetration. When a bullet hits, it transfers energy to the armor, distributed radially through longitudinal and transverse waves.

- **Longitudinal Waves:** These waves move parallel to the fibers of the armor's fabric, distributing energy efficiently along the length. This movement allows the armor to absorb energy and prevent immediate penetration by thinning and stretching the fibers.

- **Transverse Waves:** These waves move perpendicular and slower compared to longitudinal waves. They play a crucial role in damping the energy as the dent grows, moving the armor material radially and reducing the momentum of the bullet.

Energy spread effectively in the armor helps in preventing damage to the wearer. This concept aids in armor design, ensuring the kinetic energy from the impact is absorbed and spread over a broader area.

Longitudinal and Transverse Waves

When a projectile hits body armor, the impact generates two critical types of waves: longitudinal and transverse.

- **Longitudinal Waves:** These waves travel through the fabric of the armor at high speeds, responsible for quickly spreading the initial energy. They cause a significant radial inward flow of material into a formed conical dent.

- **Transverse Waves:** These waves move at a slower pace and are mainly responsible for taking care of the resultant deformation. While the longitudinal waves spread energy along the fiber length, transverse waves handle energy distribution across, increasing the depth of the impact.

Understanding these waves is crucial as they dictate how armor interacts with different projectile impacts. Designers can enhance protective gear by considering the effects of these wave dynamics, allowing designs that offer maximum protection with minimal added bulk.

Deformation Mechanics

Deformation mechanics explores how the armor material changes shape upon absorbing the energy from a projectile impact. This process involves the material's ability to withstand forces without breaking or losing structural integrity.

The conical dent formed in the fabric during impact is a perfect example of deformation. The material thins and stretches, yet returns to its initial state post impact, assuming the armor isn't permanently damaged.

- **Elastic Deformation:** Initially, the body armor can undergo elastic deformation, where the shape changes but returns to normal when the force is removed.
- **Plastic Deformation:** In extreme cases, the armor may undergo plastic deformation, where the changes are permanent. Understanding this helps in designing armor that deforms predictably and returns energy effectively, minimizing the risk of penetration and ensuring safety.