Problem 112 - Fundamentals of Physics

Question

The speed of a bullet is measured to be \(640 \mathrm{~m} / \mathrm{s}\) as the bullct cmerges from a barrel of length \(1.20 \mathrm{~m}\). A ssuming constant accelcration, find the time that the bullet spends in the barrel after it is fired.

Step-by-Step Solution

Answer

The bullet spends approximately 0.00375 seconds in the barrel.

Step 1: Identify the Known Variables

We have the initial velocity of the bullet as it starts moving in the barrel, which is \(u = 0 \ \text{m/s}\), because it starts from rest. The final velocity \(v\) is given as \(640 \ \text{m/s}\). The length of the barrel, which is the distance \(s\), is \(1.20 \ \text{m}\). We need to find the time \(t\) that the bullet spends in the barrel.

Step 2: Use the Kinematic Equation

The kinematic equation we use here is \(v^2 = u^2 + 2as\), where \(a\) is the acceleration. Since \(u = 0\), the equation simplifies to \(v^2 = 2as\). Plugging in the known values, we have \((640)^2 = 2a \times 1.20\).

Step 3: Solve for Acceleration \(a\)

Rearrange the equation from Step 2 to solve for \(a\): \[ a = \frac{(640)^2}{2 \times 1.20} \]Calculate \(a\) to get \(a = 170666.67 \ \text{m/s}^2\).

Step 4: Use the Time Equation

Now, use the kinematic equation \(v = u + at\) to solve for \(t\). Since \(u = 0\), it simplifies to \(v = at\), so \(t = \frac{v}{a}\).

Step 5: Solve for Time \(t\)

Substitute the value of \(v\) and \(a\) into the equation from Step 4:\[ t = \frac{640}{170666.67} \]Calculate \(t\) to find that the time the bullet spends in the barrel is approximately \(0.00375 \ \text{seconds}\).

Key Concepts

Constant Acceleration

Constant acceleration refers to a situation where the rate of change of velocity is uniform during the entire motion. This is a crucial concept in understanding many physical phenomena, especially in kinematics. In this context, when a bullet moves through a barrel and emerges at a known speed of 640 m/s, we assume it experiences a constant force over the barrel's length.
Understanding constant acceleration helps simplify our analysis of motion. It allows us to use specific kinematic equations that relate acceleration, initial velocity, final velocity, distance, and time.
There are some key points to keep in mind:

• Kinematic equations are only valid under constant acceleration conditions.
• By knowing the final speed and the acceleration, we can backtrack to find other unknowns, such as time.
• In our scenario, since the bullet starts from rest, its initial velocity is 0 m/s.

Bullet Speed

The speed of a bullet as it exits the barrel is a fascinating measurement that can provide insight into the gun's power and design. In our exercise, the bullet reaches a high speed of 640 m/s over a short distance of 1.20 m. This speed is determined by its acceleration throughout the barrel.
Bullet speed matters because it influences:

• Accuracy and precision: Faster speeds often yield straighter, more predictable trajectories.
• Impact force: Greater speed translates to more kinetic energy upon impact.

From a kinematic standpoint, bullet speed connects closely with acceleration. By understanding how speed grows uniformly (under constant acceleration), we can solve related problems easily, like finding the time spent in motion.

Motion Analysis

Motion analysis is the process of examining the movement of objects using fundamental physics principles. In this exercise, we analyze the bullet's motion using kinematic equations, which provide a framework for calculating vital variables like time, distance, speed, and acceleration.
Steps to conduct motion analysis:

• Identify known variables: For instance, initial velocity, final velocity, and distance.
• Choose suitable equations: Kinematic equations are applied based on the known and unknown variables.
• Solve for the unknowns: These might include finding time or acceleration as the bullet travels.

Through this structured analysis, we determine that the bullet's time inside the barrel is approximately 0.00375 seconds. By applying motion analysis techniques, complicated problems become more manageable, allowing for accurate predictions and solutions.