Problem 40 - Physics for Scientists and Engineers with Modern Physics

Question

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

Step-by-Step Solution

Answer

122,836 Joules of energy were dissipated by friction.

Step 1: Define the Problem

We are asked to find how much energy was dissipated by friction as the skier moves down the trail. This involves understanding how potential energy is converted to kinetic energy and accounting for energy lost to friction.

Step 2: Calculate Initial Potential Energy

The initial potential energy (PE) of the skier is given by the formula \( PE = mgh \), where \( m \) is mass (56 kg), \( g \) is acceleration due to gravity (9.8 m/s²), and \( h \) is the height drop (230 m). Calculate \( PE = 56 \times 9.8 \times 230 = 126,224 \) Joules.

Step 3: Calculate Final Kinetic Energy

The final kinetic energy (KE) of the skier at the bottom is calculated using the formula \( KE = 0.5mv^2 \). Here, \( m \) is 56 kg and \( v \) is 11.0 m/s. Calculate \( KE = 0.5 \times 56 \times (11)^2 = 3,388 \) Joules.

Step 4: Determine Energy Dissipated by Friction

The energy dissipated by friction is found by subtracting the final kinetic energy from the initial potential energy. Thus, the energy dissipated by friction is \( 126,224 - 3,388 = 122,836 \) Joules.

Key Concepts

Potential Energy

Potential energy is a form of stored energy that depends on an object's position relative to a reference point. For a skier at the top of a hill, it represents the energy available to be transformed as the skier descends. The formula to calculate gravitational potential energy is - \( PE = mgh \) - where:

• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity, typically \( 9.8 \, \text{m/s}^2 \) on Earth,
• \( h \) is the height of the object above the reference point.

For example, with a 56-kg skier starting at a 230 m inclination, the skier's initial potential energy is calculated as \( 126,224 \) Joules. This energy is "stored" because of the skier's elevated position and is ready to be converted into other forms of energy as the skier moves downhill.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When a skier descends, their potential energy is converted to kinetic energy. The formula for kinetic energy is - \( KE = \frac{1}{2}mv^2 \) - Where:

• \( m \) is the mass,
• \( v \) is the velocity of the object.

As the skier reaches the bottom of the hill with a speed of 11 m/s, we can calculate the final kinetic energy to be approximately \( 3,388 \) Joules. This demonstrates how much energy has been transferred from potential energy into motion, allowing the skier to glide smoothly at the base of the trail.

Friction in Physics

Friction is a force that opposes the relative motion between two surfaces in contact. It is an important concept in physics that helps explain real-world phenomena, such as the decrease in mechanical energy during movement. As a skier travels down a slope, not all potential energy converts into kinetic energy due to the work done against frictional forces. - The skier’s energy dissipation indicates the energy lost to friction. It's calculated by subtracting the final kinetic energy from the initial potential energy. In this exercise:

• Initial Potential Energy = \( 126,224 \) Joules
• Final Kinetic Energy = \( 3,388 \) Joules
• Energy dissipated by Friction = \( 126,224 - 3,388 = 122,836 \) Joules

This illustrates that a significant portion of energy is spent overcoming friction, allowing for practical examples like skiing where balance between velocity and control is essential. Understanding friction’s role in energy dissipation helps explain why not all energy translated from potential to kinetic gets you further or faster.