Penetration of a Bullet into a Wooden Block: An Analysis of Kinetic Energy and Work-Energy Principle

The interaction of a bullet with a wooden block is a fascinating example of physics in action, involving the conversion of kinetic energy into work against a resisting force. This article delves into the mathematical analysis of how far a 33 g bullet, traveling at 410 m/s, will penetrate a block of wood exerting a force of 50,000 N. By leveraging the work-energy principle, we can precisely determine the depth of penetration under these conditions.

Initial Setup and Mathematical Foundations

The problem at hand requires us to calculate the distance a bullet will travel before stopping when it strikes a block of wood. We will utilize the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.

Step 1: Calculating Initial Kinetic Energy

The first step is to calculate the initial kinetic energy of the bullet. The formula for kinetic energy is:

KE 0.5 * m * v^2

Where:

m 33 g 0.033 kg (converting grams to kilograms) v 410 m/s

Substituting the values, we get:

KE 0.5 * 0.033 * (410^2) 0.0165 * 168100 ≈ 2776.65 J

Step 2: Calculating the Work Done by the Block of Wood

The work done by the block of wood can be calculated as:

W F * d

Where:

F 50,000 N (the opposing force exerted by the block) d is the distance penetrated by the bullet, which we are trying to find.

According to the work-energy principle, the work done by the block of wood will be equal to the initial kinetic energy of the bullet. Therefore:

50,000 N * d 2776.65 J

Step 3: Solving for the Distance Penetrated

To find the distance d, we rearrange the equation:

d 2776.65 J / 50,000 N ≈ 0.05553 m 55.53 mm

This calculation shows that the bullet will penetrate approximately 55.53 mm into the block of wood.

Alternative Method

A second approach to solving the problem involves using the formula for the final velocity of the bullet in the block:

V Vo - at Vo - F/m * t

Where:

Vo 410 m/s (initial velocity) m 33 g 0.033 kg F 50,000 N

The bullet stops at t mVo/F, so the displacement (penetration distance) can be calculated as:

S Vot - 0.5 * at^2

Substituting t mVo/F, we get:

D mVo^2 / 2F

Substituting the values:

D (0.033 * 410^2) / (2 * 50,000) 0.0554 m 5.54 cm

Thus, the bullet will penetrate the block of wood at a distance of 5.54 cm.

Conclusion

By applying the work-energy principle, we have demonstrated that the bullet will penetrate approximately 55.53 mm into the block of wood, with an alternative calculation yielding 5.54 cm. These results provide insight into the interplay of kinetic energy and work in the context of bullet penetration.