Problem 16 - College Physics

Question

A gun is fired and a \(50-\mathrm{g}\) bullet is accelerated to a muzzle speed of \(100 \mathrm{~m} / \mathrm{s}\). If the length of the gun barrel is \(0.90 \mathrm{~m}\), what is the magnitude of the accelerating force? (Assume the acceleration to be constant.)

Step-by-Step Solution

Answer

The accelerating force is approximately 277.78 N.

Step 1: Analyze the given information

The problem provides the mass of the bullet as 50 grams. We must convert this to kilograms for consistency in SI units, which gives us 0.05 kg. The muzzle speed (final velocity) is given as 100 m/s, and the distance in which this acceleration occurs is 0.90 m.

Step 2: Apply the kinematic equation

To find the acceleration, we'll use the kinematic equation: \[ v^2 = u^2 + 2as \]where \( v = 100 \ m/s \) is the final velocity, \( u = 0 \ m/s \) is the initial velocity (since it starts from rest), and \( s = 0.90 \ m \) is the distance. Solving for \( a \), the equation becomes:\[ a = \frac{v^2}{2s} \]

Step 3: Calculate the acceleration

Substitute the given values into the equation:\[ a = \frac{100^2}{2 \times 0.90} = \frac{10000}{1.8} \approx 5555.56 \ m/s^2 \]Thus, the acceleration is approximately 5555.56 m/s².

Step 4: Use Newton's Second Law to find the force

Newton's Second Law states that \( F = ma \), where \( m \) is the mass in kilograms and \( a \) is the acceleration. Substituting the values:\[ F = 0.05 \times 5555.56 \approx 277.78 \ N \]

Step 5: Conclude the magnitude of the force

The magnitude of the accelerating force on the bullet is approximately 277.78 N.

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental principle in classical mechanics. It explains how the motion of an object is changed by the forces acting on it. The law is typically written as \( F = ma \), where \( F \) is the force applied to an object, \( m \) is the mass of the object, and \( a \) is the acceleration produced. This law tells us that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

When a force is applied to a bullet in a gun barrel, such as in our exercise, Newton's Second Law helps us calculate the acceleration and the force needed to launch it at a specific speed. In this scenario, given the bullet's mass and the calculated acceleration, we can use \( F = ma \) to find the force exerted.

• The bullet's mass was given as 0.05 kg.
• An acceleration of about 5555.56 m/s² was calculated based on kinematic principles.

By multiplying these values, we found that the force exerted was approximately 277.78 N.

Acceleration

Acceleration is the rate at which an object's velocity changes over time. In the context of the problem, the bullet's acceleration is a measure of how quickly it goes from being stationary to moving at its muzzle speed.

The kinematic equation \( v^2 = u^2 + 2as \) helps determine this acceleration. Here, \( v \) is the final velocity (100 m/s), \( u \) is the initial velocity (0 m/s, since the bullet starts at rest), and \( s \) is the distance over which the acceleration occurs (0.9 m).

• Rearrange this formula to solve for \( a \): \( a = \frac{v^2 - u^2}{2s} \).
• Substituting known values gives us \( a = \frac{100^2}{2 \times 0.90} \).
• This results in an acceleration of about 5555.56 m/s².

Acceleration tells us how quickly the bullet reaches its final speed as it travels through the barrel.

Muzzle Speed

Muzzle speed is the term used to describe the speed of a projectile at the moment it exits the muzzle of a gun. It's crucial for understanding the projectile's performance and behavior after leaving the barrel.

In the provided exercise, the muzzle speed is 100 m/s. This is the bullet's velocity as soon as it exits the gun's barrel, and it's achieved due to the constant acceleration it experiences over a distance of 0.9 meters.

• A high muzzle speed implies the bullet leaves the barrel with a significant amount of kinetic energy.
• This speed is pivotal for determining the bullet's impact force and range.

Understanding muzzle speed is essential for applications related to ballistics and safety. The calculations involving muzzle speed, along with concepts like acceleration and force, illustrate how physics principles apply to real-life scenarios such as firearm operation.