Textbook Question
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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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 29
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?
Verified step by step guidance1
Step 1: Identify the type of energy transformation occurring in the problem. The bicycle's initial kinetic energy is partially converted into gravitational potential energy as it climbs the ramp. Use the principle of conservation of mechanical energy to solve the problem.
Step 2: Write the equation for conservation of mechanical energy: \( KE_{initial} + PE_{initial} = KE_{final} + PE_{final} \). Here, \( KE \) represents kinetic energy \( \frac{1}{2}mv^2 \), and \( PE \) represents gravitational potential energy \( mgh \).
Step 3: Simplify the equation by noting that \( PE_{initial} \) is zero because the ramp starts at ground level. The equation becomes \( \frac{1}{2}mv_{initial}^2 = \frac{1}{2}mv_{final}^2 + mgh \). Cancel out the mass \( m \) from all terms since it appears in every term.
Step 4: Rearrange the equation to solve for \( v_{final} \), the speed of the bicycle at the top of the ramp: \( v_{final} = \sqrt{v_{initial}^2 - 2gh} \). Substitute the given values: \( v_{initial} = 8.0 \ \text{m/s} \), \( g = 9.8 \ \text{m/s}^2 \), and \( h = 1.0 \ \text{m} \).
Step 5: Perform the substitution and simplify the expression for \( v_{final} \). This will give the bicycle's speed as it leaves the top of the ramp. Ensure units are consistent throughout the calculation.
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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 bicycle's initial kinetic energy will be converted into potential energy as it climbs the ramp, and then back into kinetic energy as it descends. This concept is crucial for determining the speed of the bicycle at the top of the ramp.
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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. As the bicycle coasts up the ramp, its kinetic energy decreases as it converts to potential energy. Understanding this relationship is essential for solving the problem.
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06:07Intro to Rotational Kinetic Energy
Potential Energy
Potential energy is the stored energy of an object due to its position in a gravitational field, calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is height. As the bicycle ascends the ramp, it gains potential energy, which affects its speed at the top. This concept is vital for analyzing the energy transformations involved.
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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
FIGURE EX2.32 shows the acceleration graph for a particle that starts from rest at t = 0 s. What is the particle's velocity at t = 6 s?
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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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