91Ó°ÊÓ

To determine the average acceleration experienced by an object, we must understand the concept of change in velocity over a specific period of time. Average acceleration is defined as the rate at which velocity changes. It's denoted by the formula:

• \[ a = \frac{\Delta v}{t} \]
• Where \(\Delta v\) is the change in velocity and \(t\) is the time taken.

In the context of our bullet problem, the bullet's velocity changes from zero (since we assume it starts from rest) to \(262 \, \mathrm{m/s}\) as it travels through a \(5.0\)-inch barrel in \(1 \, \mathrm{ms}\) (or \(0.001 \, \mathrm{s}\)). The substitution of these values gives us an average acceleration of \(262,000 \, \mathrm{m/s^2}\). This high value is typical for such rapid change in speed over a short time span, as experienced in firearms.

Newton's Second Law is a core principle in physics, describing how the velocity of an object changes when it is subjected to an external force. This law is succinctly expressed by the equation:

• \[ F = ma \]
• where \(F\) is the force applied, \(m\) is the mass of the object, and \(a\) is the acceleration.

This is especially pertinent in calculating the force on a bullet as it is accelerated within the barrel. When we know the mass of the bullet (\(0.0162 \, \mathrm{kg}\)) and its acceleration (\(262,000 \, \mathrm{m/s^2}\)), we can compute the force using Newton's second law. This results in a force of \(4,244.4 \, \mathrm{N}\). This immense force is what propels the bullet at such a high speed out of the gun.

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause the motion. In the context of our problem, we focus on the bullet's movement within the barrel, described by its initial and final velocities, the time it takes to traverse the barrel, and its acceleration. The key parameters include:

• Initial Velocity: Often \(0 \, \mathrm{m/s}\) when starting from rest
• Final Velocity: \(262 \, \mathrm{m/s}\) for our bullet
• Time Span: \(0.001 \, \mathrm{s}\), the time taken to cover the barrel length
• Acceleration: derived from change in velocity \(\Delta v\) over time \(t\)

These aspects define the bullet's motion as it speeds up from the back to the front of the gun barrel, allowing for the calculation of acceleration and force.

Force calculation, as derived from the principles of Newton's Second Law, is crucial for understanding how fast and powerfully a bullet exits a firearm. Using the formula:

• \[ F = ma \]

We first determine the bullet's acceleration using kinematics principles, and then apply it to calculate the force. The mass of the bullet is \(0.0162 \, \mathrm{kg}\), and the acceleration is \(262,000 \, \mathrm{m/s^2}\) as computed previously. Thus, the average force exerted on the bullet inside the gun is \(4,244.4 \, \mathrm{N}\). This showcases how small objects can exert powerful impacts due to high acceleration, emphasizing the power of physics in real-world applications.