Problem 40 - Physics for Scientists and Engineers with Modern Physics

Question

(1I) A 56 -kg skier starts from rest at the top of a 1200 -m- long trail which drops a total of 230 m from top to bottom. At the bottom, the skier is moving 11.0 \(\mathrm{m} / \mathrm{s} .\) How much. energy was dissignated by friction?

Step-by-Step Solution

Answer

The energy dissipated by friction is 122668 J.

Step 1: Calculate Potential Energy at the Top

The potential energy (PE) at the top of the hill is given by the formula \( PE = mgh \), where \( m = 56 \) kg is the mass of the skier, \( g = 9.8 \) m/s² is the acceleration due to gravity, and \( h = 230 \) m is the height of the hill. Substituting these values, we find \( PE = 56 \times 9.8 \times 230 \).

Step 2: Evaluate the Total Initial Energy

Since the skier starts from rest, the initial kinetic energy (KE) is 0. Therefore, the total initial energy is just the potential energy calculated in Step 1, which is \( PE = 126056 \) J.

Step 3: Calculate Kinetic Energy at the Bottom

The kinetic energy (KE) at the bottom of the hill is given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m = 56 \) kg and \( v = 11.0 \) m/s. Substituting these values, we find \( KE = \frac{1}{2} \times 56 \times (11.0)^2 \), which calculates to \( KE = 3388 \) J.

Step 4: Determine Total Work Done by Friction

The work done by friction is the difference between the initial potential energy and the final kinetic energy. It is given by \( W = PE - KE \). Substituting \( PE = 126056 \) J and \( KE = 3388 \) J, we calculate \( W = 126056 - 3388 \) J, which results in \( W = 122668 \) J.

Key Concepts

Potential Energy

Potential energy is the energy an object possesses because of its position relative to other objects. In physics, this is usually related to its position in a gravitational field. For the skier at the top of the hill, potential energy can be understood as the stored energy due to the elevated position on the slope.
When the skier is at the top, the potential energy is calculated using the formula:

• \( PE = mgh \)

where \( m \) is the mass of the skier (in kilograms), \( g \) represents the acceleration due to gravity (approximately \( 9.8 \) m/s² on the surface of the Earth), and \( h \) stands for the height from which the skier is starting. In this exercise, the skier's potential energy is determined by multiplying the mass (56 kg), gravitational constant (9.8 m/s²), and the height (230 m). This results in a potential energy of 126056 joules (J) at the start.
Understanding potential energy helps us predict how much energy is available to convert into other forms, such as kinetic energy, as the skier descends the trail.

Kinetic Energy

Kinetic energy refers to the energy an object has due to its motion. When the skier reaches the bottom of the hill, potential energy has transformed primarily into kinetic energy as the skier moves at a certain speed.
The kinetic energy at the bottom of the hill is determined using the formula:

• \( KE = \frac{1}{2} mv^2 \)

In this case, the mass \( m \) remains the same, at 56 kg, and the speed \( v \) is 11.0 m/s. Substituting these values into the formula, you calculate the kinetic energy, which comes out to be 3388 joules (J).
This change in energy showcases how potential energy is converted to kinetic energy as the skier moves down the hill. The motion at the bottom indicates not just a transition of energy forms, but an overall understanding of how energy flow works under the principles of physics.

Work Done by Friction

Even though energy naturally transitions from potential to kinetic forms, friction plays a significant role by dissipating some of the energy. Work done by friction is an important concept in this context and is responsible for energy loss as heat, sound, or deformation.
The work done by friction is calculated by finding the difference between the total initial potential energy and the kinetic energy at the bottom of the hill:

• \( W = PE - KE \)

Here, PE is the potential energy (126056 J) and KE is the kinetic energy (3388 J), giving us a work done by friction value of 122668 joules (J).
Understanding the work done by friction helps us comprehend energy conservation in real-world scenarios. This concept ensures that all energy transformations, including losses, align with the law of conservation of energy, ensuring no energy is unaccounted for in an isolated system.