Problem 8 - College Physics

Question

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 1.74 m/s. (b) 179.6 J of kinetic energy is created, sourced from the skaters' internal energy.

Step 1: Understand the Problem

We have two ice skaters with different weights who push off each other on frictionless ice. We need to find the speed of the lighter skater after the push and calculate the increase in kinetic energy that results from their interaction.

Step 2: Apply the Law of Conservation of Momentum

Since the ice is frictionless, the momentum of the system is conserved. The total momentum before the skaters push each other is zero (since both start at rest), so it remains zero after they push off.Let the velocity of the heavier skater be \(v_{h} = 1.50 \mathrm{~m/s}\), and let the velocity of the lighter skater be \(v_{l}\).Express the conservation of momentum as:\[ m_{h} \cdot v_{h} = m_{l} \cdot v_{l} \]Where:- \( m_{h} = \frac{725\: \mathrm{N}}{9.81\: \mathrm{m/s^2}} \)- \( m_{l} = \frac{625\: \mathrm{N}}{9.81\: \mathrm{m/s^2}} \)

Step 3: Solve for the Velocity of the Lighter Skater

Using the conservation of momentum equation:\[ \left(\frac{725}{9.81}\right) \times 1.50 = \left(\frac{625}{9.81}\right) \times v_{l} \]Solve for \(v_{l}\):\[ v_{l} = \frac{725 \times 1.50}{625} = 1.74\: \mathrm{m/s} \]

Step 4: Calculate Initial Kinetic Energy

Calculate the initial kinetic energy, which is zero because both skaters are initially at rest:\[ KE_{initial} = 0 \]

Step 5: Calculate Final Kinetic Energy

Calculate the final kinetic energy for each skater and sum them:For the heavier skater:\[ KE_{h} = \frac{1}{2} m_{h} v_{h}^2 = \frac{1}{2} \left(\frac{725}{9.81}\right) 1.50^2 = 82.8 \: \mathrm{J} \]For the lighter skater:\[ KE_{l} = \frac{1}{2} m_{l} v_{l}^2 = \frac{1}{2} \left(\frac{625}{9.81}\right) 1.74^2 = 96.8 \: \mathrm{J} \]Total kinetic energy:\[ KE_{final} = 82.8 + 96.8 = 179.6 \: \mathrm{J} \]

Step 6: Determine the Energy Creation and Source

The change in kinetic energy is the difference between final and initial kinetic energy:\[ \Delta KE = KE_{final} - KE_{initial} = 179.6 \: \mathrm{J} \]This energy comes from the internal energy of the skaters, as they exert force to push off each other.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When the figure skaters push off from each other, they gain velocity, and thus their kinetic energy increases. The heavier skater travels at a speed of 1.50 m/s, while the lighter skater achieves a velocity calculated to be 1.74 m/s.
To find the kinetic energy for each skater, we use the formula: \[ KE = \frac{1}{2} mv^2 \] - For the heavier skater, the kinetic energy turns out to be approximately 82.8 J (joules). - For the lighter skater, it's approximately 96.8 J.
Initially, when both skaters were at rest, their kinetic energy was zero. After they push off, the total kinetic energy becomes 179.6 J. This increase is directly due to their acquired velocities.

Frictionless Surface

The scenario takes place on a frictionless ice surface, which is crucial for the conservation principles at play. A frictionless surface ensures that no external forces such as friction oppose the motion of the skaters.
This setting implies certain conditions:

• There is no energy loss due to frictional forces.
• The net external force on the system is zero.
• This allows us to apply the law of conservation of momentum perfectly.

For students, keeping in mind the uniqueness of a frictionless environment aids in understanding why the total momentum is conserved so seamlessly here.

Momentum Conservation

Momentum conservation is a fundamental principle in physics stating that the total momentum of a closed system remains constant, provided no external forces are acting on it. In the case of the skaters, we start with zero initial momentum because both were at rest.
When they push off, each skater moves in opposite directions. The conservation of momentum can be expressed with the equation: \[ m_{h}v_{h} = m_{l}v_{l} \] where:

• \( m_{h} \) and \( m_{l} \) are the masses of the heavier and lighter skater, respectively.
• \( v_{h} \) and \( v_{l} \) are their velocities.

Since momentum before and after the event must be equal, we set the product of the heavier skater's mass and velocity equal to that of the lighter skater, facilitating the calculation for the lighter skater's velocity.

Internal Energy Source

The energy used by the skaters to push off each other originates from their internal energy sources. This includes muscular energy, which they convert into mechanical energy during the maneuver.
When they exert force against each other, internal energy is transformed:

• The coupled force of their push generates the motion.
• This action creates the kinetic energy observed in both skaters.

The increase in kinetic energy—179.6 J—is essentially the energy produced by their muscles, showcasing how living organisms can generate energy internally to perform tasks.