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Chapter 7: Problem 29

\(\mathrm{A} 62.0 \mathrm{~kg}\) skier is moving at \(6.50 \mathrm{~m} / \mathrm{s}\) on a frictionless, horizontal, snow-covered plateau when she encounters a rough patch \(4.20 \mathrm{~m}\) long. The coefficient of kinetic friction between this patch and her skis is 0.300 . After crossing the rough patch and returning to friction-free snow, she skis down an icy, frictionless hill \(2.50 \mathrm{~m}\) high. (a) How fast is the skier moving when she gets to the bottom of the hill? (b) How much internal energy was generated in crossing the rough patch?

Short Answer

Expert verified
The speed of the skier at the bottom of the hill can be calculated using the conservation of energy principle, which considers both her initial kinetic energy (reduced by the work done by friction while crossing the rough patch) and her potential energy at the top of the hill. The internal energy generated when crossing the rough patch is equal to the work done by friction.

Step by step solution

01

Calculate the work done by friction

First, we calculate the work done by friction when the skier crosses the rough patch. The work done is given by the formula: \( Work = Force \times Distance \). The force of friction is \( 碌 \times mass \times gravity \), where \( 碌 \) is the coefficient of friction (0.300), the mass is 62.0 kg and the gravitational force on earth is approximately \( 9.8 m/s^2 \). The distance the skier covers in the rough path is 4.20 m.
02

Calculate the initial kinetic energy

Next, we calculate the initial kinetic energy of the skier before reaching the rough patch using the formula: \( kinetic \, energy = 1/2 \times mass \times speed^2 \), where the mass is 62.0 kg and the initial speed is 6.50 m/s.
03

Calculate the final kinetic energy

The final kinetic energy will be the difference between the initial kinetic energy and the work done by friction. The speed of the skier after crossing the rough patch can be calculated by rearranging the kinetic energy formula: \( speed = sqrt ((2 \times kinetic \, energy / mass )) \).
04

Calculate the speed at the bottom of the hill

After crossing the rough patch, the skier skis down an icy, frictionless hill which is 2.50 m high. We can calculate her speed at the bottom of the hill by applying the principle of conservation of energy. At the top of the hill, she has both kinetic energy (calculated in the previous step) and potential energy due to the height of the hill. At the bottom of the hill, all this energy will have been converted into kinetic energy. The potential energy at the top of the hill is given by the formula: \( potential \, energy = mass \times gravity \times height \), where the height is 2.50 m. The total kinetic energy at the bottom of the hill will be the sum of the potential energy and the kinetic energy at the top of the hill. Her speed at the bottom of the hill can then be calculated using the kinetic energy formula.
05

Calculate the internal energy generated

The internal energy generated when the skier crossed the rough patch is equal to the work done by friction.

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Key Concepts

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

Friction
Friction is a force that opposes the motion of objects. In the context of our skier, friction occurs when she hits the rough patch of snow. Despite the surrounding region being frictionless, the rough patch creates resistance. This force is pivotal as it generates work done against the skier's motion.

The work done by friction can be calculated using the formula:
  • Work = Force of Friction x Distance
Force of Friction is derived from:
  • Force = coefficient of friction (渭) x mass x gravitational acceleration
Knowing the skier's mass (62.0 kg), the friction coefficient (0.300), and gravity (approximately 9.8 m/s虏), we can calculate the work done over the 4.20 m rough path. This work done by friction directly affects her kinetic energy, slowing her down before she reaches the hill.
Conservation of Energy
The conservation of energy principle states that the total energy in a closed system remains constant. It merely transforms from one type to another. As the skier passes over the rough patch, the kinetic energy due to her motion gets partially converted into internal energy due to friction. Once she descends the hill, potential energy at the top (due to her height) converts back into kinetic energy as she reaches the bottom.

In this case:
  • The skier starts with kinetic energy gained from her initial speed.
  • Upon hitting the rough patch, some of this energy is transformed into work done by friction.
  • As she ascends to the top of the hill, her kinetic energy decreases, and potential energy increases.
  • Descending the hill, potential energy is reconverted to kinetic energy, thereby increasing her speed.
Potential Energy
Potential energy is the stored energy of an object due to its position. For the skier, this is the energy associated with her elevation on the hill. As she reaches the top, her potential energy is at a maximum. This potential energy contributes to her speed increase as she descends.

The expression used to calculate potential energy is:
  • Potential Energy = mass x gravitational force x height
Given the skier's mass (62.0 kg), gravitational force (9.8 m/s虏), and the hill's height (2.50 m), we can compute the potential energy at the top. This energy, crucially, is converted into kinetic energy as she descends, enhancing her descent speed.
Work-Energy Principle
The work-energy principle directly ties the concept of work to changes in energy. Specifically, it states that the work done by all forces acting on an object will result in a change in its kinetic energy. For the skier, as she moves over the rough patch, the work done by friction reduces her kinetic energy.

The process unfolds as follows:
  • Initial kinetic energy is calculated from her initial speed.
  • The rough patch introduces a frictional force doing work, which decreases her kinetic energy by this exact amount.
  • The resulting kinetic energy after the patch is less due to this energy loss (work done).
Understanding how work and kinetic energy interplay helps in predicting speed variations due to energy transformations.

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