Problem 29 - University Physics with Modern Physics

Question

A 62.0-kg skier is moving at 6.50 m/s on a frictionless, horizontal, snow- covered plateau when she encounters a rough patch 4.20 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 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?

Step-by-Step Solution

Answer

(a) 8.16 m/s at the bottom of the hill. (b) 765.58 J of internal energy.

Step 1: Calculate Deceleration on Rough Patch

The force of kinetic friction acting on the skier is given by the formula \( f_k = \mu_k \cdot m \cdot g \), where \( \mu_k = 0.300 \), \( m = 62.0 \; \text{kg} \), and \( g = 9.8 \; \text{m/s}^2 \). So, \( f_k = 0.300 \times 62.0 \times 9.8 = 182.28 \; \text{N} \). The acceleration due to friction is \( a = \frac{f_k}{m} = \frac{182.28}{62.0} = 2.94 \; \text{m/s}^2 \), acting in the opposite direction of motion.

Step 2: Determine Final Velocity after Rough Patch

Using the equation of motion \( v^2 = u^2 + 2a s \), where \( u = 6.50 \; \text{m/s} \), \( a = -2.94 \; \text{m/s}^2 \), and \( s = 4.20 \; \text{m} \), we find the final velocity. \( v^2 = (6.50)^2 + 2 \times (-2.94) \times 4.20 = 42.25 - 24.7 = 17.55 \). Thus, \( v = \sqrt{17.55} = 4.19 \; \text{m/s} \).

Step 3: Calculate Velocity at Bottom of Hill

When the skier descends the 2.50 m hill, potential energy converts to kinetic energy. The change in potential energy is \( mgh = 62.0 \times 9.8 \times 2.50 = 1519 \; \text{J} \). The change in kinetic energy equals the change in potential energy. The initial kinetic energy at the top of the hill is \( \frac{1}{2} \times 62.0 \times (4.19)^2 = 544.64 \; \text{J} \). At the bottom, \( \text{KE}_{\text{bottom}} = \text{KE}_{\text{top}} + \text{PE change} = 544.64 + 1519 = 2063.64 \; \text{J} \). Solve \( \frac{1}{2} \times 62.0 \times v^2 = 2063.64 \) for \( v \), giving \( v^2 = \frac{2063.64 \times 2}{62.0} = 66.57 \), so \( v = 8.16 \; \text{m/s} \).

Step 4: Calculate Internal Energy Generated

The internal energy generated is equal to the work done by friction over the rough patch, given by \( W = f_k \times s = 182.28 \times 4.20 = 765.58 \; \text{J} \). This is the energy converted to heat due to friction.

Key Concepts

Frictional Force

Frictional force is a resistive force that opposes the motion or attempted motion of an object on a surface. It's important to understand that friction can be both a help and a hindrance in physics problems. In the case of our skier, this concept is illustrated when she encounters a rough patch on her journey. The force of kinetic friction, which is the type that acts on moving objects, plays a critical role here. It is calculated using the formula:

• \( f_k = \mu_k \cdot m \cdot g \)

Here, \( \mu_k \) is the coefficient of kinetic friction, \( m \) is the mass, and \( g \) is the acceleration due to gravity. In our problem, this frictional force reduces the skier's speed as she travels across the rough patch, turning kinetic energy into heat and, therefore, affecting her overall motion.

Potential Energy

Potential energy is the stored energy of an object based on its position relative to a reference point, usually the ground. For our skier, she has potential energy at the top of the hill after crossing the rough patch. This energy can be calculated using:

• \( PE = mgh \)

Where \( m \) is mass, \( g \) is gravitational acceleration (\(9.8 \; \text{m/s}^2\)), and \( h \) is the height above the ground. In her descent down a 2.50 m high hill, this potential energy is converted into kinetic energy, increasing her speed as she reaches the bottom. This transformation is an essential part of the concept of energy conservation.

Kinetic Energy

Kinetic energy is the energy of motion. Whenever an object is moving, it has kinetic energy, which can be calculated using:

• \( KE = \frac{1}{2}mv^2 \)

In the skier's problem, she starts with an initial kinetic energy as she encounters the rough patch. However, due to the deceleration caused by friction, some of this kinetic energy is converted to heat, causing her speed to decrease. This calculation helps determine how her speed is affected by friction and later how it is affected by the potential energy she gains as she climbs the hill, which then converts back to kinetic energy as she descends.

Energy Conservation

Energy conservation is a fundamental concept in physics stating that the total energy of an isolated system remains constant. In the skier's scenario, energy conservation helps us track how energy is transferred and transformed from one form to another. Initially, the skier has kinetic energy which is partially lost to friction as she crosses the rough patch. When she reaches the hill, her position gives her potential energy. As she descends, this potential energy transforms back into kinetic energy. This principle allows us to determine her velocity at the bottom of the hill by understanding these energy transformations.

Physics Problem Solving

Physics problem solving often involves applying a series of logical steps to address a given problem. In our skier's exercise, this includes:

• Identifying the forces involved, such as friction, which is tackled by calculating the frictional force and its impact on the skier's speed.
• Recognizing energy forms, such as potential and kinetic energy, and how they convert as the skier moves.
• Applying the laws of motion and energy conservation to solve for unknowns, like the skier's speed at different points of her journey.
• Calculating work done by forces, like friction, to determine how much energy is lost or transformed.

These steps guide us through the problem logically, ensuring that each aspect of the skier's movement and energy changes is thoroughly understood and addressed.