Problem 43 - College Physics

Question

A boy of mass mo and his girlfriend of mass \(m\), both wearing ice skates, face each other at rest while standing on a frictionless joe rink. The boy pushes the girl, sending her away with velocity \(\tau_{k}\) toward the east. Assume that \(m_{b}>m_{e}\). (a) Describe the subsequent motion of the boy. (b) Find expressions for the final kinetic energy of the girl and the Final kinetic energy of the boy, and show that the girl has greater kinetic energy than the boy. (c) The boy and girl had zero kinetic energy before the boy pushed the girl, but ended up with kinetic energy after the event. How do you account for the appearance of mechanical energy?

Step-by-Step Solution

Answer

After the push, the boy moves in the opposite direction to the girl with a lower velocity. The kinetic energy of the girl ends up being greater than that of the boy by a factor of the ratio of their masses squared. This change in energy can be explained by the work done by the boy on the girl when he pushed her.

Step 1: Conservation of Momentum

Using the principle of conservation of momentum, since initially they both were at rest, the total initial momentum is zero. After the boy pushes the girl, the total final momentum still should be zero. The final momentum can be represented mathematically by the equation: \(0 = m_{boy} * v_{boy} - m_{girl} * v_{girl}\). Here, \(v_{boy}\) is the velocity of the boy and \(v_{girl}\) is the velocity of the girl. From this, we can solve for \(v_{boy}\), the boy's velocity, which is \(v_{boy} = (m_{girl}/m_{boy}) * v_{girl}\). Equivalent to \(v_{boy} = \tau_{k} * (m_{e}/m_{b})\) .

Step 2: Kinetic Energy

After finding \(v_{boy}\), we can find the kinetic energy of both the boy and the girl using the formula for kinetic energy \(K = 1/2 * m * v^2\). Substituting the masses and the velocities we obtain, the kinetic energy of the girl is \(K_{girl} = 1/2 * m * \tau_{k}^2\), and the kinetic energy of the boy is \(K_{boy} = 1/2 * m_{boy} * (\tau_{k} * m_{e}/m_{b})^2\). Due to initial condition we know that \(m_{b}>m_{e}\), hence, \(K_{girl} > K_{boy}\).

Step 3: Mechanical Energy

The total mechanical energy of the system was initially zero, but increased due to the work done by the boy, which we can account for through the work-energy theorem. This concept states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the boy in pushing the girl turned into the kinetic energy for both the boy and the girl.

Key Concepts

Kinetic Energy

In physics, **kinetic energy** is the energy an object possesses due to its motion. It is dependent on two main factors: the mass of the object and its velocity. The formula to calculate kinetic energy is quite simple: \[ K = \frac{1}{2} m v^2 \] where \( K \) represents kinetic energy, \( m \) is the mass, and \( v \) is the velocity of the object.

• If an object is not moving, its kinetic energy is zero.
• The faster an object moves, the greater its kinetic energy.
• For two objects with the same velocity, the one with greater mass will have more kinetic energy.

In the context of the original exercise, we see how kinetic energy plays out in a real-world scenario: when the boy pushes the girl, both gain kinetic energy from being initially stationary. Given the girl's speed and mass, her kinetic energy is more significant than the boy's, given the same initial work exertion but different masses.

Frictionless Surface

A **frictionless surface** is an idealized concept where there is no resistance to an object's motion across the surface. In essence, it means that once an object starts moving on such a surface, it will continue to move forever without slowing down.

• Real-world examples include ice or nearly friction-free air tracks used in physics labs.
• Without friction, external forces are the only way to change the motion of an object on such surfaces.

In the exercise, the "frictionless joe rink" ensures that once the boy and girl begin to move, they will not lose speed due to friction. This means the only factors influencing their velocities are their initial push and their masses. Thus, it perfectly showcases the conservation of momentum and demonstrates motion without energy dissipation due to friction.

Work-Energy Theorem

The **work-energy theorem** is a fundamental principle that connects the work done on an object to its kinetic energy. It states that the work done by all forces acting on an object is equal to the change in the object's kinetic energy:\[ W = \Delta K \]where \( W \) is the work done, and \( \Delta K \) is the change in kinetic energy.

• The theorem simplifies the process of analyzing a dynamic system where forces are at play.
• In the exercise's context, the work done by the boy on the girl is what changes their kinetic energies from zero to their respective values after the push.

The increase in total mechanical energy observed after the event is a direct consequence of the work-energy theorem. The boy exerted a force over a certain distance, accomplishing work that manifested as increased kinetic energy for both the boy and the girl. This explains why they both gain speed and move across the ice.

Newton's Third Law

**Newton's Third Law** of motion is famously summarized as "for every action, there is an equal and opposite reaction." This law is a cornerstone of classical mechanics and dictates that forces always occur in pairs. When one body exerts a force on another, the second body exerts an equal but opposite force back on the first body. For our ice-skating pair:

• The force the boy exerts on the girl during the push creates an equal and opposite reaction force exerted by the girl on the boy.
• This explains why both the boy and the girl move in opposite directions after the push.

This reciprocal action enhances the understanding of conservation of momentum and how each participant in the system influences the other's motion. Therefore, after the push, both the boy and girlfriend move apart, demonstrating the intrinsic balance of forces where, although they both generate kinetic energy, their momentum remains conserved as per the total system.