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Chapter 8: Problem 15

The drawing shows a device that can be used to measure the speed of a bullet. The device consists of two rotating disks, separated by a distance of \(d=0.850 \mathrm{m},\) and rotating with an angular speed of 95.0 rad/s. The bullet first passes through the left disk and then through the right disk. It is found that the angular displacement between the two bullet holes is \(\theta=0.240\) rad. From these data, determine the speed of the bullet.

Short Answer

Expert verified
The speed of the bullet is approximately 336.47 m/s.

Step by step solution

01

Understand the Problem

The device measures the speed of a bullet by using two rotating disks with a known angular speed. The bullet passes through both disks, and the angular displacement between the bullet holes provides the key information needed to determine the bullet's speed.
02

Establish Known Variables

The distance between the two disks is \(d = 0.850\, \text{m}\), the angular speed of the disks is \(\omega = 95.0\, \text{rad/s}\), and the angular displacement between the two bullet holes is \(\theta = 0.240\, \text{rad}\).
03

Calculate the Time taken between the Disks

The time \(t\) taken for the bullet to travel from the first to the second disk can be calculated using the relation between angular speed \(\omega\), angular displacement \(\theta\), and time: \(\theta = \omega \times t\). Rearrange this to solve for \(t\): \(t = \frac{\theta}{\omega}\). Substitute the given values: \(t = \frac{0.240 \, \text{rad}}{95.0 \, \text{rad/s}} = 0.0025263 \, \text{s}\).
04

Calculate the Speed of the Bullet

Use the formula relating speed \(v\), distance \(d\), and time \(t\): \(v = \frac{d}{t}\). Substitute the known values to find the speed of the bullet: \(v = \frac{0.850 \, \text{m}}{0.0025263 \, \text{s}} \approx 336.47 \, \text{m/s}\). Ensure units are consistent and the calculation is correct.

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Key Concepts

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

Understanding Angular Displacement
Angular displacement refers to the angle in radians through which a point or line has been rotated in a specified direction around a specific axis. It is a measure of how much rotation has occurred between two different points in time. In this bullet speed measurement device, the angular displacement is the angle between the bullet holes as the bullet passes through two rotating disks.
  • Angular displacement is denoted by 胃 and is measured in radians.
  • It is a scalar quantity, meaning it only has magnitude, not direction.
  • In this context, 胃 = 0.240 rad, indicating how much the first disk has turned relative to the second when the bullet passes through both.
When solving such problems, recognizing that the angular displacement helps relate the bullet's speed to its position at the exit of each disk is crucial.
Conceptualizing Angular Speed
Angular speed is a measure of how quickly an object rotates or revolves relative to another point. It's how fast the angular displacement changes over time and is often represented by the symbol 蠅 (omega). Here, angular speed lets you understand how fast the disks are spinning.
  • Angular speed is measured in radians per second (rad/s).
  • In this problem, 蠅 = 95.0 rad/s, which tells us that each disk completes 95 radians of rotation in one second.
  • It is a vector quantity, often depicted in physics with a direction using the right-hand rule, but for calculations like this, magnitude is key.
By understanding angular speed, one can connect it with angular displacement to solve for time, which is crucial for determining speeds in these types of physics problems.
Role of Rotating Disks in Bullet Speed Measurement
Rotating disks in this device are essential for calculating the bullet's speed. The disks are spaced apart by a known distance, and they rotate with a known angular speed.
  • The layout with separation "d = 0.850 m" between the disks allows for calculation of travel time for the bullet across this distance, under rotation.
  • The rotation introduces observable points - punch holes - which are critical for measuring angular displacement.
  • This angular displacement between the two bullet holes aids in formulating a relationship with time using the equation 胃 = 蠅 脳 t.
Understanding the function that these rotating disks serve is vital when using angular mechanics to solve problems involving speeds of fast-moving objects like bullets.
Mastering Physics Problem-Solving
Physics problem-solving often involves breaking down a problem into manageable steps, which can help in gaining a clear understanding of the involved principles and applying them effectively. For this problem:
  • Step 1: Comprehend the problem's setup鈥攌nowing what each variable represents and what you ultimately need to find.
  • Step 2: Identify known and unknown variables. Here, known variables include 蠅, d, and 胃.
  • Step 3: Use equations that connect the given information, such as 胃 = 蠅 脳 t, to find any unknowns, like the travel time t.
  • Step 4: Plug your findings into the speed formula v = d/t to solve for the desired value, which is the bullet speed in this scenario.
Following these structured steps helps ensure all aspects of the problem are covered, and calculations are carried out correctly, making physics less daunting and more logical.

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Most popular questions from this chapter

An automobile tire has a radius of \(0.330 \mathrm{m},\) and its center moves forward with a linear speed of \(v=15.0 \mathrm{m} / \mathrm{s}\) (a) Determine the angular speed of the wheel. (b) Relative to the axle, what is the tangential speed of a point located \(0.175 \mathrm{m}\) from the axle?

A 220 -kg speedboat is negotiating a circular turn (radius \(=32 \mathrm{m})\) around a buoy. During the turn, the engine causes a net tangential force of magnitude \(550 \mathrm{N}\) to be applied to the boat. The initial tangential speed of the boat going into the turn is \(5.0 \mathrm{m} / \mathrm{s}\). (a) Find the tangential acceleration. (b) After the boat is \(2.0 \mathrm{s}\) into the turn, find the centripetal acceleration.

The drawing shows a chain-saw blade. The rotating sprocket tip at the end of the guide bar has a radius of \(4.0 \times 10^{-2} \mathrm{m} .\) The linear speed of a chain link at point \(\mathrm{A}\) is \(5.6 \mathrm{m} / \mathrm{s}\). Find the angular speed of the sprocket tip in rev/s.

Two people start at the same place and walk around a circular lake in opposite directions. One walks with an angular speed of \(1.7 \times 10^{-3} \mathrm{rad} / \mathrm{s}\), while the other has an angular speed of \(3.4 \times 10^{-3} \mathrm{rad} / \mathrm{s} .\) How long will it be before they meet?

A top is a toy that is made to spin on its pointed end by pulling on a string wrapped around the body of the top. The string has a length of \(64 \mathrm{cm}\) and is wound around the top at a spot where its radius is \(2.0 \mathrm{cm}\). The thickness of the string is negligible. The top is initially at rest. Someone pulls the free end of the string, thereby unwinding it and giving the top an angular acceleration of \(+12 \mathrm{rad} / \mathrm{s}^{2} .\) What is the final angular velocity of the top when the string is completely unwound?
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