Textbook Question
Below what speed does a 3.0-mm-diameter ball bearing in 20°C air experience linear drag?
Ch 06: Dynamics I: Motion Along a Line
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 6, Problem 30
A 50,000 kg locomotive is traveling at 10 m/s when its engine and brakes both fail. How far will the locomotive roll before it comes to a stop? Assume the track is level.
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Identify the given values: The mass of the locomotive \( m = 50,000 \, \text{kg} \), the initial velocity \( v_i = 10 \; \text{m/s} \), and the final velocity \( v_f = 0 \; \text{m/s} \). Since the engine and brakes fail, the stopping force is due to friction, but the coefficient of friction is not provided. Assume the deceleration is constant and use kinematic equations.
Write the kinematic equation that relates initial velocity, final velocity, acceleration, and displacement: \( v_f^2 = v_i^2 + 2a d \), where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, \( a \) is the acceleration, and \( d \) is the displacement (distance traveled).
Rearrange the equation to solve for displacement \( d \): \( d = \frac{v_f^2 - v_i^2}{2a} \). Since \( v_f = 0 \), the equation simplifies to \( d = -\frac{v_i^2}{2a} \).
Determine the acceleration \( a \): The acceleration is caused by the frictional force, which can be expressed as \( F_f = \mu m g \), where \( \mu \) is the coefficient of friction and \( g \) is the acceleration due to gravity. The acceleration \( a \) is then \( a = -\mu g \). Substitute this into the displacement equation: \( d = \frac{v_i^2}{2 \mu g} \).
To find the distance \( d \), you need the coefficient of friction \( \mu \). If \( \mu \) is provided or assumed, substitute its value along with \( v_i = 10 \; \text{m/s} \) and \( g = 9.8 \; \text{m/s}^2 \) into the equation \( d = \frac{v_i^2}{2 \mu g} \) to calculate the distance.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Kinematics
Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this scenario, we need to analyze the locomotive's motion as it decelerates to a stop, which can be described using kinematic equations.
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Friction
Friction is the force that opposes the relative motion of two surfaces in contact. In this case, the locomotive will experience kinetic friction between its wheels and the track, which will decelerate it until it comes to a stop. The coefficient of friction is crucial for calculating the distance traveled before stopping.
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Energy Conservation
The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the locomotive's kinetic energy will be converted into thermal energy due to friction as it rolls to a stop. Understanding this concept helps in calculating the distance based on the initial kinetic energy and the work done by friction.
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Related Practice
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A 1500 kg car skids to a halt on a wet road where = 0.50. How fast was the car traveling if it leaves 65-m-long skid marks?
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