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A bullet traveling horizontally at a speed of \(350 \mathrm{~m} / \mathrm{s}\) hits a board perpendicular to its surface, passes through and emerges on the other side at a speed of \(210 \mathrm{~m} / \mathrm{s}\). If the board is \(4.00 \mathrm{~cm}\) thick, how long does the bullet take to pass through it?

Short Answer

Expert verified
The bullet takes approximately \(1.43 \times 10^{-4} \, \text{s}\) to pass through the board.

Step by step solution

01

Understand the Problem

We are given a bullet traveling horizontally with an initial speed of \(350 \, \text{m/s}\) and emerging with a final speed of \(210 \, \text{m/s}\) after passing through a board that is \(4.00 \, \text{cm}\) thick. We need to find the time it takes for the bullet to pass through the board.
02

Convert Units

Convert the thickness of the board from centimeters to meters to match the units of speed. \[ 4.00 \, \text{cm} = 0.04 \, \text{m} \]
03

Calculate Average Speed

The average speed of the bullet while passing through the board is the mean of the initial and final velocities. Use the formula: \[ v_{\text{avg}} = \frac{v_i + v_f}{2} \] where \(v_i = 350 \, \text{m/s}\) and \(v_f = 210 \, \text{m/s}\). Calculating this gives: \[ v_{\text{avg}} = \frac{350 + 210}{2} = 280 \, \text{m/s} \]
04

Use Average Speed to Calculate Time

The time \(t\) it takes to travel a distance at an average speed is given by the formula: \[ t = \frac{d}{v_{\text{avg}}} \] where \(d = 0.04 \, \text{m}\) and \(v_{\text{avg}} = 280 \, \text{m/s}\). Calculating this gives: \[ t = \frac{0.04}{280} \approx 1.43 \times 10^{-4} \, \text{s} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Physics Problems
Physics problems, like the one involving the bullet and the board, help us understand how objects interact with forces and matter in the real world. In this problem, the bullet travels horizontally, which adds a layer of complexity. We are required to think about the bullet's speed before and after it penetrates the board, and how this helps in calculating the time it takes to pass through. These problems encourage critical thinking by requiring us to convert units and apply formulas accurately. Here's how we can break down the key components of physics problems:
  • Identify given values and what's being asked.
  • Ensure all units are compatible.
  • Apply relevant physics laws and formulas.
This problem represents a classic scenario of applying basic physics concepts to a seemingly simple question, revealing the underlying physics principles at play.
Kinematics
Kinematics is the branch of physics focused on motion, without regard to forces. It helps us describe how objects move. In this problem, we're dealing with linear motion, which falls under kinematics. The bullet is initially at a certain speed, travels through the board, and exits at another speed. Key aspects of kinematics in this context involve:
  • Velocity: The bullet has an initial velocity and a final velocity, which are essential to compute the average velocity in the situation.
  • Displacement: We know the thickness of the board is the total displacement the bullet undergoes along its path through the board.
Understanding kinematics enables us to compute these values and apply them to calculate other quantities, such as time, accurately.
Velocity and Speed
Velocity and speed are related but distinct concepts in physics. **Velocity** is a vector quantity, which means it has both a magnitude and a direction, while **speed** is a scalar quantity and only has magnitude. In our exercise:
  • The bullet's **initial velocity** is 350 m/s, and its **final velocity** is 210 m/s, both horizontally directed.
  • The **average speed** is calculated as the arithmetic mean of these two velocities, which forms the basis for understanding the bullet's motion through the board.
The focus is on the speed, as it affects how quickly the bullet travels. By knowing its speed before and after hitting the board, we can calculate the average speed and then the time taken to travel through a known distance.
Time Calculation
Time calculation is crucial in determining how long processes take, using known values like speed and distance. To find how long the bullet takes to pass through the board, we use the formula for time given by kinematics:
  • The **formula** is \( t = \frac{d}{v_{\text{avg}}} \), where \( t \) is time, \( d \) is the distance, and \( v_{\text{avg}} \) is the average speed.
  • In this problem, \( d = 0.04 \) meters (after converting from centimeters) and \( v_{\text{avg}} \) is calculated to be 280 m/s.
By dividing the distance by the average speed, we can find the time the bullet spends passing through the board, which is approximately 1.43 脳 10鈦烩伌 seconds. Mastering this calculation is vital in solving many physics problems involving movement over a period.

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Most popular questions from this chapter

You throw a stone vertically upward with an initial speed of \(6.0 \mathrm{~m} / \mathrm{s}\) from a third-story office window. If the window is \(12 \mathrm{~m}\) above the ground, find (a) the time the stone is in flight and (b) the speed of the stone just before it hits the ground.

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