Problem 8 - College Physics

Question

\(\cdot\) Two figure skaters, one weighing 625 \(\mathrm{N}\) and the other \(725 \mathrm{N},\) push off against each other on frictionless ice. (a) If the heavier skater travels at 1.50 \(\mathrm{m} / \mathrm{s}\) , how fast will the lighter one travel? (b) How much kinetic energy is "created" during the skaters' maneuver, and where does this energy come from?

Step-by-Step Solution

Answer

(a) The lighter skater travels at approximately 1.738 m/s. (b) 179.31 Joules of kinetic energy is created from the skaters' muscular energy.

Step 1: Understand the Problem

In this problem, two figure skaters push off against each other on frictionless ice. We need to determine the speed of the lighter skater after the push and find out the kinetic energy created during the maneuver. This involves the laws of conservation of momentum and energy.

Step 2: Determine the Masses of the Skaters

Convert the weights of the skaters to masses. Since weight \( W = mg \), where \( g = 9.8 \, \text{m/s}^2 \), calculate:1. Heavier skater's mass: \( m_2 = \frac{725}{9.8} \, \approx 74 \, \text{kg} \).2. Lighter skater's mass: \( m_1 = \frac{625}{9.8} \, \approx 63.78 \, \text{kg} \).

Step 3: Apply Conservation of Momentum

The momentum of the system is conserved. Before the push, total momentum is zero, so after the push, it remains zero.\[ m_1 v_1 = m_2 v_2 \]where \( v_2 = 1.5 \, \text{m/s} \).Solve for \( v_1 \):\[ v_1 = \frac{m_2 v_2}{m_1} = \frac{74 \times 1.5}{63.78} \approx 1.738 \, \text{m/s} \].

Step 4: Compute Kinetic Energy Created

The kinetic energy of each skater after the push is given by \( KE = \frac{1}{2} mv^2 \).1. Heavier skater: \( KE_2 = \frac{1}{2} \times 74 \times 1.5^2 = 83.25 \, \text{Joules} \).2. Lighter skater: \( KE_1 = \frac{1}{2} \times 63.78 \times 1.738^2 \approx 96.06 \, \text{Joules} \).Total kinetic energy created: \( KE = 83.25 + 96.06 \approx 179.31 \, \text{Joules} \).

Step 5: Analyze the Source of Energy

The energy comes from the internal muscular energy exerted by the skaters during the push. This is converted into kinetic energy, increasing their speed on ice.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. For any object in motion, kinetic energy can be calculated using the formula: \[ KE = \frac{1}{2} mv^2 \] where:

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

In the context of the skaters, once they push off against each other, they both gain speed. Their kinetic energy is the result of the conversion of their muscle energy into motion energy.
Each skater's kinetic energy is calculated individually and then summed to find the total kinetic energy created in the system during the maneuver. This transformation demonstrates how energy changes form but is conserved within the system.

Frictionless Ice

Frictionless surfaces provide an ideal condition where no resistance opposes the movement of objects. On such surfaces, any force exerted leads to motion without any energy lost to friction.
In our exercise, the ice is considered frictionless, which simplifies the problem. It means that energy isn't leaked out as heat due to friction, making calculations more straightforward.

• Momentum conservation becomes evident because no external forces are acting on the skaters other than their initial push.
• Kinetic energy calculations directly reflect the energy exerted by the skaters without reduction by friction.

This "slippery" nature makes ice a great place to study pure momentum and energy interactions in physics exercises.

Conservation of Energy

The principle of conservation of energy states that energy in an isolated system remains constant - it cannot be created or destroyed, only transformed. In this exercise, the energy is initially stored as potential energy in the muscles of the skaters.
When they push off each other, the energy is transformed into kinetic energy.

• The total energy before and after must remain the same, thereby explaining the conversion process that takes place.
• Even in systems where different forms of energy interact, the total energy remains unchanged unless acted upon by external forces.

This principle is a cornerstone concept of physics, ensuring that all calculations align with the natural laws governing energy transformations.