Calculating the Average Resistive Force Exerted by a Wooden Plank on a Moving Bullet

Introductory Explanation

In this detailed article, we will delve into the process of calculating the average resistive force exerted by a wooden plank on a bullet. By utilizing the work-energy principle, we will derive the necessary equations and values to determine the resistance encountered by the bullet as it penetrates the wooden plank. This analysis is crucial for understanding the dynamics between the bullet and the material it impacts.

Understanding the Problem

Consider a bullet with a mass of 20 grams (0.02 kg) moving with a velocity of 75 m/s. Upon hitting a fixed wooden plank, it comes to rest after penetrating a distance of 5 cm (0.05 m).

Step-by-Step Calculation

Step 1: Calculate the Initial Kinetic Energy of the Bullet

First, we calculate the initial kinetic energy (KE) of the bullet using the formula:

KE (1/2)mv2

Where:

m 0.02 kg (mass of the bullet) v 75 m/s (speed of the bullet)

Plugging these values into the formula:

KE (1/2) × 0.02 × 752

KE 0.01 × 5625

KE 56.25 J

Step 2: Calculate the Work Done by the Resistive Force

The work done (W) by the resistive force is given by:

W Fd

Where:

F average resistive force (what we want to find) d 0.05 m (penetration distance)

Since the bullet comes to rest, the work done by the resistive force is equal to the negative change in kinetic energy:

W -KE -56.25 J

Step 3: Solve for the Average Resistive Force

Substitute the values into the work equation:

-56.25 J F × 0.05 m

Solving for F:

F -56.25 J / 0.05 m

F -1125 N

Since force is a vector quantity, we will use the positive value (magnitude) of 1125 N. Thus, the average resistive force exerted by the wooden plank on the bullet is approximately 1125 N in the direction opposite to the bullet's motion.

Alternative Method for Verification

Let's verify the calculation using the alternative method provided:

The initial velocity (u) is 90 m/s and the final velocity (v) is 0 m/s. The distance (s) is 0.05 m. The formula to find the acceleration (a) is:

Vf2 Vo2 2as

Rearranging to solve for ‘a’:

a (Vf2 - Vo2) / 2s

Substituting the values:

a (0 - 902) / 0.1

a -81000 / 0.1

a -810000 m/s2

The force (F) using Newton's second law (F ma) is:

F 0.04 kg × (-810000 m/s2)

F -32400 N

The force calculated using this method is much higher than the earlier result. This discrepancy may be due to the incorrect interpretation of the given values in the alternative solution or the context of the problem. However, the primary method discussed here yields a more realistic value.

Conclusion

Upon analysis, the average resistive force exerted by the wooden plank on the bullet is approximately 1125 N. This force acts in the direction opposite to the bullet's motion, effectively decelerating and stopping the bullet as it penetrates the wooden plank.

References:

Knight, D. (2014). Physics for Scientists and Engineers. Pearson. Halliday, D., Resnick, R., Walker, J. (2014). Fundamentals of Physics. Wiley.

Keywords:average resistive force, resistance calculation, bullet physics