Problem 40 - College Physics

Question

A 270 caliber hunting rifle fires an 8.5 g bullet, which exits the gun barrel at a speed of \(900 \mathrm{~m} / \mathrm{s}\). (a) What impulse does the burning gunpowder impart to the bullet? (b) If it takes \(2 \mathrm{~ms}\) for the bullet to travel the length of the barrel, what is the average force on the bullet? Express your answer in pounds.

Step-by-Step Solution

Answer

(a) Impulse is 7.65 kg·m/s. (b) Average force is 859.84 lb.

Step 1: Identify the known quantities

We are given that the mass of the bullet is 8.5 grams, which we convert to kilograms: 0.0085 kg. The speed of the bullet as it exits the barrel is 900 m/s. The time taken for the bullet to travel the length of the barrel is 2 ms, which we convert to seconds: 0.002 s.

Step 2: Calculate the impulse imparted to the bullet

Impulse is defined as the change in momentum, which can also be calculated as the product of the average force and the time period. However, momentum is mass times velocity: \( \text{Impulse} = m \times v \), where \( m = 0.0085 \text{ kg} \) and \( v = 900 \text{ m/s} \). Substitute the values to get \( \text{Impulse} = 0.0085 \times 900 = 7.65 \text{ kg} \cdot \text{m/s} \).

Step 3: Calculate the average force on the bullet

The impulse can also be expressed as \( \text{Impulse} = F_{\text{avg}} \times \Delta t \), where \( F_{\text{avg}} \) is the average force and \( \Delta t \) is time. Rearrange to find \( F_{\text{avg}} = \frac{\text{Impulse}}{\Delta t} = \frac{7.65}{0.002} = 3825 \text{ N} \).

Step 4: Convert force from newtons to pounds

We know that 1 N = 0.224809 lb. Use this conversion to convert the force from newtons to pounds: \( 3825 \text{ N} \times 0.224809 = 859.84 \text{ lb} \).

Key Concepts

Momentum

Momentum in physics describes the quantity of motion an object possesses. It's a fundamental concept when studying motion, especially in relation to forces and impulse. Momentum ( )p ) is calculated as the product of an object's mass ( )m ) and its velocity ( )v ): \[ p = m imes v \] In the given exercise, the bullet's momentum can be determined by multiplying its mass (converted to kilograms as 0.0085 kg) by its velocity (900 m/s). The result is a momentum of 7.65 kg·m/s. Understanding momentum is essential because it helps us describe and predict various interactions and changes a moving object may undergo. For example, the momentum of a bullet gives insight into its movement through the air after being fired. It helps us solve problems related to collisions and is used widely in physics beyond just mechanics.

Average Force Calculation

Calculating average force involves understanding how forces change an object's momentum over a time period. Impulse, which equates to change in momentum, is defined as: \[ \text{Impulse} = F_{\text{avg}} \times \Delta t \] Where \( F_{\text{avg}} \) is the average force applied, and \( \Delta t \) is the time interval. In our exercise, impulse was found to be 7.65 kg·m/s. With a given time of 0.002 seconds, we rearrange the formula to find average force: \[ F_{\text{avg}} = \frac{\text{Impulse}}{\Delta t} = \frac{7.65}{0.002} = 3825 \text{ N} \] This calculation reveals the transient force exerted on the bullet as it travels through the barrel, giving an idea of the power involved in firing the bullet, which in real-world scenarios could be vital for understanding recoil and gun mechanics.

Unit Conversion

Understanding and using unit conversion is key in physics to ensure calculations are accurate and results are interpretable. In this exercise, we perform two conversions: mass and force.

• Mass: The bullet's mass is given in grams, a common unit of measure in chemistry and biology. For physics calculations, however, converting this to kilograms is essential, because the fundamental unit of mass in SI units is kilograms. For example, 8.5 grams converts to 0.0085 kg.
• Force: The calculated force of 3825 N needs conversion into pounds for context in regions where imperial units are prevalent. Using the conversion factor 1 N = 0.224809 lb, the force was converted to roughly 859.84 lb.

Correct unit conversion enhances communication cross-discipline and ensures mathematics underlying scientific principles remains consistent across varied measurement systems.