Problem 38 - Physics

Question

In \(10.0\) s, 200 bullets strike and embed themselves in a wall. The bullets strike the wall perpendicularly. Each bullet has a mass of \(5.0 \times 10^{-3} \mathrm{~kg}\) and a speed of \(1200 \mathrm{~m} / \mathrm{s}\). (a) What is the average change in momentum per second for the bullets? (b) Determine the average force exerted on the wall. (c) Assuming the bullets are spread out over an area of \(3.0 \times 10^{-4} \mathrm{~m}^{2}\), obtain the average pressure they exert on this region of the wall.

Step-by-Step Solution

Answer

(a) 120 kg m/s², (b) 120 N, (c) 4.0 x 10⁵ Pa.

Step 1: Calculate the momentum change for one bullet

The momentum change for one bullet, assuming it comes to rest after embedding in the wall, is given by the initial momentum of the bullet. Momentum (\( p \)) is defined as mass (\( m \)) times velocity (\( v \)).Given: \[ m = 5.0 \times 10^{-3} \text{ kg}, \quad v = 1200 \text{ m/s} \]We calculate:\[ p = m \times v = 5.0 \times 10^{-3} \times 1200 = 6.0 \text{ kg m/s} \]So, the change in momentum for one bullet is \(6.0 \text{ kg m/s}\).

Step 2: Calculate total momentum change for 200 bullets

Since each bullet has a momentum change of \(6.0 \text{ kg m/s}\), the total momentum change for \(200\) bullets is calculated by multiplying the momentum change of one bullet by the total number of bullets: \[ \text{Total Change in Momentum} = 200 \times 6.0 = 1200 \text{ kg m/s} \]

Step 3: Determine average change in momentum per second

The average change in momentum per second (rate of change of momentum) is found by dividing the total change in momentum by the time (\(10.0\) s):\[ \text{Average Change in Momentum per Second} = \frac{1200}{10.0} = 120 \text{ kg m/s}^2 \]

Step 4: Calculate the average force exerted on the wall

The average force exerted on the wall is equal to the average change in momentum per second, by the definition of force in classical physics (\( F = \frac{\Delta p}{\Delta t} \)): \[ F = 120 \text{ N} \]

Step 5: Determine the average pressure exerted on the wall

Pressure (\( P \)) is defined as the force (\( F \)) applied per unit area (\( A \)). Given that the bullets contact an area of \(3.0 \times 10^{-4} \text{ m}^2\), calculate pressure: \[ P = \frac{F}{A} = \frac{120}{3.0 \times 10^{-4}} = 4.0 \times 10^{5} \text{ Pa} \]

Key Concepts

Change in Momentum

The concept of change in momentum is crucial when analyzing collisions and impacts. In the context of our exercise, each bullet impacts the wall and comes to a stop, resulting in a complete loss of its initial momentum. Momentum is determined by the formula: \[ p = m \times v \] Where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. For a single bullet, with a mass of \(5.0 \times 10^{-3} \text{ kg}\) and a velocity of \(1200 \text{ m/s}\), the momentum is calculated as: \[ p = 5.0 \times 10^{-3} \times 1200 = 6.0 \text{ kg m/s} \]The change in momentum for one bullet is equal to this value, \(6.0 \text{ kg m/s}\), as it comes to a stop. This calculation helps us understand how each bullet contributes to the overall change in momentum when 200 bullets strike the wall. To find this overall change in momentum for all bullets, we simply multiply the change in momentum of one bullet by the total number of bullets: \[ \text{Total Change in Momentum} = 200 \times 6.0 = 1200 \text{ kg m/s} \] Understanding change in momentum allows us to measure how the effect of multiple impacts combines over time.

Average Force

Average force is a fundamental concept in physics, representing how an overall force is distributed over a specific time period. In this exercise, the definition of average force helps us determine how the momentum change over time is translated into force. According to Newton's second law, force is the rate of change of momentum, expressed as: \[ F = \frac{\Delta p}{\Delta t} \] Here, \( \Delta p \) is the total change in momentum, and \( \Delta t \) is the time interval. We have calculated the total change in momentum as \(1200 \text{ kg m/s}\) over a \(10.0\) second time period. Thus, the average force exerted on the wall can be found as follows: \[ F = \frac{1200}{10} = 120 \text{ N} \] This implies that the wall experiences a constant force of \(120\) newtons due to the continued impact of the bullets over the given time span. This calculation is essential in understanding how forces interact with surfaces over time, especially in engineering and safety applications.

Pressure Calculation

Pressure is a measure of how much force is applied over a particular area. It's particularly significant in scenarios involving impacts over surfaces, as in our exercise where bullets hit a wall. The pressure exerted by a force can be calculated using the formula: \[ P = \frac{F}{A} \] In this equation, \( P \) is pressure, \( F \) is force, and \( A \) is the area over which the force is distributed. Given that the average force exerted on the wall is \(120 \text{ N}\) and this force is spread out over an area of \(3.0 \times 10^{-4} \text{ m}^2\), the pressure applied to this area is calculated as: \[ P = \frac{120}{3.0 \times 10^{-4}} = 4.0 \times 10^{5} \text{ Pa} \] This calculation shows that the bullets exert a pressure of \(4.0 \times 10^{5} \text{ pascals (Pa)}\) on the wall. Understanding pressure is critical in predicting how materials respond under various forces, which is vital for designing sustainable and safe structures.