Textbook Question
A bicycle coasting at 8.0 m/s comes to a 5.0-m-long, 1.0-m-high ramp. What is the bicycle's speed as it leaves the top of the ramp?
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Ch 02: Kinematics in One Dimension
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 2, Problem 27a
A skier is gliding along at 3.0 m/s on horizontal, frictionless snow. He suddenly starts down a 10° incline. His speed at the bottom is 15 m/s. What is the length of the incline?
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Identify the known quantities: initial velocity \( v_i = 3.0 \; \text{m/s} \), final velocity \( v_f = 15.0 \; \text{m/s} \), and the angle of the incline \( \theta = 10^\circ \). The goal is to find the length of the incline \( d \).
Use the work-energy principle, which states that the change in kinetic energy is equal to the work done by gravity. The equation is \( \Delta KE = W_g \), where \( \Delta KE = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 \) and \( W_g = m g d \sin(\theta) \).
Cancel out the mass \( m \) from both sides of the equation since it appears in every term. This simplifies to \( \frac{1}{2} v_f^2 - \frac{1}{2} v_i^2 = g d \sin(\theta) \).
Rearrange the equation to solve for \( d \): \( d = \frac{v_f^2 - v_i^2}{2 g \sin(\theta)} \).
Substitute the known values into the equation: \( v_f = 15.0 \; \text{m/s} \), \( v_i = 3.0 \; \text{m/s} \), \( g = 9.8 \; \text{m/s}^2 \), and \( \sin(10^\circ) \). Perform the calculations to find \( d \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conservation of Energy
The principle of conservation of energy states that in a closed system, the total energy remains constant. In this scenario, the skier's initial kinetic energy and potential energy at the top of the incline will convert into kinetic energy at the bottom. This concept is crucial for determining the relationship between the skier's speeds and the height of the incline.
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06:24
Conservation Of Mechanical Energy
Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In this problem, the skier's speed changes from 3.0 m/s to 15 m/s, indicating a change in kinetic energy as he descends the incline, which is essential for calculating the length of the incline.
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06:07Intro to Rotational Kinetic Energy
Inclined Plane Dynamics
Inclined plane dynamics involves analyzing the motion of objects on slopes, taking into account gravitational forces and angles. The angle of the incline affects the component of gravitational force acting along the slope, which influences the skier's acceleration. Understanding this concept helps in determining the length of the incline based on the skier's change in speed.
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06:59Intro to Inclined Planes
Related Practice
Textbook Question
FIGURE EX2.31 shows the acceleration-versus-time graph of a particle moving along the x-axis. Its initial velocity is v0x = 8.0 m/s at t0 = 0 s. What is the particle’s velocity at t = 4.0s?
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Textbook Question
When jumping, a flea accelerates at an astounding 1000 m/s2, but over only the very short distance of 0.50 mm. If a flea jumps straight up, and if air resistance is neglected (a rather poor approximation in this situation), how high does the flea go?
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Textbook Question
As a science project, you drop a watermelon off the top of the Empire State Building, 320 m above the sidewalk. It so happens that Superman flies by at the instant you release the watermelon. Superman is headed straight down with a speed of 35 m/s. How fast is the watermelon going when it passes Superman?
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Textbook Question
A snowboarder glides down a 50-m-long, 15° hill. She then glides horizontally for 10 m before reaching a 25° upward slope. Assume the snow is frictionless. How far can she travel up the 25° slope?
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Textbook Question
A car traveling at 30 m/s runs out of gas while traveling up a 10° slope. How far up the hill will it coast before starting to roll back down?
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