Problem 84 - College Physics

Question

\(\bullet\) \(\bullet\) On an essentially frictionless horizontal ice-skating rink, a skater moving at 3.0 \(\mathrm{m} / \mathrm{s}\) encounters a rough patch that reduces her speed by 45\(\%\) due to a friction force that is 25\(\%\) of her weight. Use the work-energy principle to find the length of the rough patch.

Step-by-Step Solution

Answer

The rough patch is approximately 1.28 meters long.

Step 1: Understanding the Problem

We need to calculate the length of the rough patch based on the skater's initial speed, the reduction in speed, and the force of friction. The initial speed is 3.0 m/s and the speed is reduced by 45%, so the final speed is 3.0 m/s * 55% = 1.65 m/s. The friction force is 25% of her weight.

Step 2: Determine the Initial and Final Kinetic Energies

The initial kinetic energy (KE) is given by the formula \( KE_i = \frac{1}{2}mv_i^2 \), where \( v_i = 3.0 \text{ m/s} \). The final kinetic energy is \( KE_f = \frac{1}{2}mv_f^2 \), where \( v_f = 1.65 \text{ m/s} \). Here, \( m \) is the mass of the skater.

Step 3: Calculate the Work Done by Friction

The work-energy principle states that the work done by friction is equal to the change in kinetic energy: \( W = KE_f - KE_i \). The force of friction is \( f = 0.25mg \), where \( g \) is the acceleration due to gravity. So, \( W = fd \), where \( d \) is the length of the rough patch.

Step 4: Set Up the Work-Energy Equation

We have \( 0.25mg \cdot d = \frac{1}{2}m(1.65)^2 - \frac{1}{2}m(3.0)^2 \). Notice that the mass \( m \) cancels out, so the equation simplifies to \( 0.25g \cdot d = \frac{1}{2} \cdot (1.65)^2 - \frac{1}{2} \cdot (3.0)^2 \).

Step 5: Solve for the Length of the Rough Patch

Insert the value of \( g = 9.8 \text{ m/s}^2 \) and solve for \( d \):\[0.25 \times 9.8 \times d = \frac{1}{2}(1.65)^2 - \frac{1}{2}(3.0)^2\]Calculate each part:- \( \frac{1}{2}(1.65)^2 \approx 1.36125 \text{ J} \)- \( \frac{1}{2}(3.0)^2 = 4.5 \text{ J} \)Now solve:\[ 0.25 \times 9.8 \times d = 1.36125 - 4.5 = -3.13875 \]\[ d = \frac{-3.13875}{0.25 \times 9.8} \approx 1.28 \text{ meters} \]

Step 6: Conclusion

The length of the rough patch is approximately 1.28 meters. This means that the skater experienced a deceleration over this distance due to the friction force, reducing her speed by 45%.

Key Concepts

Kinetic Energy

In physics, kinetic energy is the energy that an object possesses due to its motion. The faster an object moves, or the more massive it is, the more kinetic energy it has. The formula to calculate kinetic energy is:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass of the object, and \( v \) is its velocity.
Kinetic energy plays a crucial role when dealing with moving objects because it helps us understand how much work an object can do due to its motion. In our skater example, the initial kinetic energy can be calculated using her initial speed of 3.0 m/s.
Once she hits the rough patch, her speed decreases to 1.65 m/s, and thus her kinetic energy decreases. This reduction in kinetic energy indicates that work is done against the force of friction, which we'll explore next.

Friction Force

Friction is the resistance that one surface or object encounters when moving over another. It is an opposing force, meaning it acts against the direction of motion. In our problem, the friction force is 25% of the skater's weight.
Mathematically, this can be expressed as:

• \( f = 0.25mg \)

where \( m \) is the mass of the skater, and \( g \) is the acceleration due to gravity (roughly 9.8 m/s² on Earth).
Frictional forces do negative work, which means they take energy away from the system. Here, the skater loses energy as she moves over the rough patch, contributing to her reduction in speed. By calculating the work done by friction, using the equation \( W = fd \), we can determine how the friction affects the skater's motion on the ice.

Acceleration Due to Gravity

Gravity is a natural phenomenon by which all things with mass or energy are brought toward one another. On the surface of the Earth, this acceleration due to gravity is approximately 9.8 m/s².
In the scenario with the skater, the acceleration due to gravity is fundamental in calculating the friction force since the skater's weight, which is the force due to gravity, determines the strength of the friction force.
Every object on Earth's surface experiences this constant, making it a key factor in any physics problem involving forces, especially those that require a calculation of weight using the formula:

• \( ext{Weight} = mg \)

This value (9.8 m/s²) is consistent around the Earth's surface and is vital when examining the work-energy principle in scenarios involving motion like our skating example.

Horizontal Motion

Horizontal motion refers to movement in a straight line parallel to the horizon. It is an important concept when analyzing events such as a skater gliding across an ice rink.
In most physics problems, especially those involving the work-energy principle, horizontal motion significantly simplifies the calculations.
This is because typically, there is no vertical motion involved, meaning gravitational potential energy does not change, and we are mainly concerned with changes in kinetic energy and the work done by forces like friction.
In the given scenario, the skater starts her journey with a certain horizontal velocity, which decreases as she encounters friction. By focusing solely on horizontal motion, we isolate only the forces acting in that direction, such as the kinetic friction force that opposes her glide across the rough patch.