Question

A bullet is fired through a board \(10.0 \mathrm{~cm}\) thick, with a line of motion perpendicular to the face of the board. If the bullet enters with a speed of \(400 . \mathrm{m} / \mathrm{s}\) and emerges with a speed of \(200 . \mathrm{m} / \mathrm{s}\), what is its acceleration as it passes through the board?

Short answer

Answer: The acceleration of the bullet is -600,000 m/s².

Step-by-step solution

Identify the given values

We are given the following: - Initial velocity (\(v_i\)): \(400\,\text{m/s}\) - Final velocity (\(v_f\)): \(200\,\text{m/s}\) - Displacement (\(x\)): \(0.1\,\text{m}\) (converted from 10 cm)

Use the formula for acceleration

To find the acceleration, we will use the equation: \(v_f^2 = v_i^2 + 2ax\)

Plug in the known values

Insert the given values into the equation: \((200\,\text{m/s})^2 = (400\,\text{m/s})^2 + 2a(0.1\,\text{m})\)

Solve for 'a'

To solve for 'a', we need to perform the following calculations: 1. Square both velocities: \((200^2\,\text{m}^2/\text{s}^2) = 40000\,\text{m}^2/\text{s}^2\) and \((400^2\,\text{m}^2/\text{s}^2) = 160000\,\text{m}^2/\text{s}^2\) 2. Subtract the squared initial velocity from the squared final velocity: \(40000 - 160000 = -120000\,\text{m}^2/\text{s}^2\) 3. Divide both sides by \(2 \times 0.1\,\text{m}\): \(a = \frac{-120000\,\text{m}^2/\text{s}^2}{2 \times 0.1\,\text{m}}\)

Calculate the acceleration

Perform the division to find the acceleration: \(a = \frac{-120000\,\text{m}^2/\text{s}^2}{0.2\,\text{m}} = -600000\,\text{m/s}^2\) Therefore, the acceleration of the bullet as it passes through the board is \(-600000\,\text{m/s}^2\).

Key concepts

Kinematics Equations

Kinematic equations are the cornerstone of classical mechanics, particularly useful for solving problems involving motion with constant acceleration. These equations relate displacement, initial and final velocities, acceleration, and time without requiring the detailed understanding of the force causing the acceleration. The equation used in our exercise is one of those powerful tools. It can be given as \( v_f^2 = v_i^2 + 2ax \), where \( v_i \) is the initial velocity, \( v_f \) is the final velocity, \( a \) is the acceleration, and \( x \) is the displacement. When we know any three of these variables, we can solve for the fourth.

It's crucial to remember that these equations apply only to uniformly accelerated motion, where the acceleration does not change over time. They are highly practical for a wide range of real-world problems, such as the bullet acceleration problem from our exercise.

Initial and Final Velocity

Understanding initial and final velocities is crucial when analyzing motion. The initial velocity, often denoted as \( v_i \), is the velocity of the object before the influence of an acceleration, while the final velocity, \( v_f \), is the velocity after acceleration has been applied. In the context of our bullet acceleration problem, the bullet's initial velocity is high as it enters the board and decreases due to the opposing force of the board, resulting in a lower final velocity as it exits. This change in velocity, along with displacement, allows us to infer the acceleration experienced by the bullet, even without knowing the exact forces at play.

Uniformly Accelerated Motion

Uniformly accelerated motion refers to motion where the acceleration, \( a \), is constant throughout the duration of the movement. This implies that the velocity changes at a steady rate. For the bullet passing through the board in our problem, we assume that it undergoes uniform acceleration, meaning the force exerted by the board on the bullet is constant while the bullet is in contact with the material. This assumption simplifies our calculations and is a common approach when dealing with short durations or distances, where variable forces may not significantly alter the acceleration.

Displacement and Acceleration Relationship

The relationship between displacement and acceleration is a key concept in kinematics. Displacement, which is the distance moved in a specific direction, is directly affected by acceleration when the motion is along a straight line, as with our bullet problem. The kinematic equation \( v_f^2 = v_i^2 + 2ax \) lets us see this relationship mathematically, showing how displacement (\( x \) in the equation) is affected by the initial and final velocities (\( v_i \) and \( v_f \)) and acceleration (\( a \)). Positive acceleration leads to an increase in the final velocity, given the initial velocity and displacement, while negative acceleration, as seen in the solved problem, reduces the final velocity for the bullet passing through the board.