Chapter 6: Problem 139
Think \& Calculate A sled slides without friction down a small, ice-covered hill. If the sled starts from rest at the top of the hill, its speed at the bottom is \(7.50 \mathrm{~m} / \mathrm{s}\). (a) On a second run, the sled starts with a speed of \(1.50 \mathrm{~m} / \mathrm{s}\) at the top. When it reaches the bottom of the hill, is its speed \(9.00 \mathrm{~m} / \mathrm{s}\), more than \(9.00 \mathrm{~m} / \mathrm{s}\), or less than \(9.00 \mathrm{~m} / \mathrm{s}\) ? Explain. (b) Find the speed of the sled at the bottom of the hill after the second run.
Short Answer
Step by step solution
Understand the Concept
Establish Formula for First Run
Establish Formula for Second Run
Simplify and Compare
Calculate the Result
Answer Part (a)
Finalize the Answer for Part (b)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Physics
The absence of friction lets us simplify the problem using the conservation of energy equation. All the potential energy from the top becomes kinetic energy at the bottom, showcasing a perfect conversion, which is a fundamental physics concept. So, physics is essentially showing us how different forms of energy are interchangeable without loss in an idealized environment.
Kinetic Energy
- \( KE = \frac{1}{2}mv^2 \)
Initially, when the sled is at the top, it may have some kinetic energy if it starts with an initial speed. As it moves down the hill, potential energy is converted to kinetic energy. At the bottom, all potential energy has become kinetic energy, thus increasing the sled's speed. This relationship highlights the kinetic energy's dependence on an object's velocity: as it speeds up, its kinetic energy increases.
For the sled starting from rest (first run), this kinetic energy at the bottom tells us how fast it travels. For the second run with a starting speed, this initial kinetic energy adds to what is gained from potential energy conversion, although still not enough to reach the speed of 9.00 m/s due to the energy constraints outlined earlier.
Potential Energy
- Height (\( h \))
- Mass of the sled (\( m \))
- Gravitational field strength (\( g \))
In a frictionless scenario, this conversion is efficient, allowing potential energy to completely transform into kinetic energy. The greater the height, the more potential energy the sled has initially. Thus, a higher hill results in a greater speed at the bottom due to more energy being converted. This exercise demonstrates, in both runs, how potential energy at the top becomes kinetic energy at the bottom without any loss.
Frictionless Motion
In reality, friction would slow down the sled, converting kinetic energy into heat, making it impossible to fully convert potential energy into kinetic energy. However, in frictionless motion, all potential energy transfers to kinetic energy without any deductions along the way. It's worth noting that setting up problems this way helps simplify complex real-world physics scenarios into more easily computable exercises.
Idealized frictionless motion enables us to use fundamental physics concepts to understand and predict motion outcomes, providing a clear example of how potential energy transforms into kinetic energy without interruptions.