Problem 48 - College Physics

Question

Identical twins, each with mass 55.0 kg, are on ice skates and at rest on a frozen lake, which may be taken as frictionless. Twin A is carrying a backpack of mass ats fractionless. Twin A is carrying a backpack of inass \(12.0 \mathrm{~kg}\). She throws it horizomtally at \(9.00 \mathrm{~m} / \mathrm{s}\) to Twin B. Neglecting any gravity effects, what are the subsequent speeds of Twin \(A\) and Twin \(B\) ?

Step-by-Step Solution

Answer

Twin A moves at a speed of 1.96 m/s in the opposite direction to the throw, and Twin B (with the backpack) moves at a speed of 1.61 m/s in the same direction as the throw.

Step 1: Calculate total mass and momentum

First, calculate the total mass of the system before the throw. This is the combined mass of Twin A (with the backpack), Twin B, and the backpack. The total momentum before the throw is 0 because both twins are at rest. So, total mass = 55 kg (Twin A) + 55 kg (Twin B) + 12 kg (backpack) = 122 kg. Since both twins were at rest, the initial momentum is 0.

Step 2: Calculate the momentum after Twin A throws the backpack

After Twin A throws the backpack, the backpack has a momentum of 12 kg * 9 m/s = 108 kgm/s. Remember, momentum = mass * velocity. Because the total momentum of this closed system must remain zero (due to the law of conservation of momentum), Twin A must then have a momentum of -108 kgm/s (in the opposite direction). You can find Twin A's velocity by dividing her momentum by her mass. Thus, Twin A's velocity = -108 kgm/s / 55 kg = -1.96 m/s.

Step 3: Calculate the speed of Twin B after he catches the backpack

When Twin B catches the backpack, they are together considered a single object with a combined mass of 55 kg (Twin B) + 12 kg (backpack) = 67 kg. After Twin B catches the backpack, the momentum of the combined Twin B and the backpack must be 108 kgm/s (opposite to the direction of Twin A) in order to keep the total momentum at 0. So, the velocity (or speed) of Twin B (after catching the backpack) is momentum / mass = 108 kgm/s / 67 kg = 1.61 m/s.

Key Concepts

Momentum Calculation

Understanding momentum is crucial when studying the dynamics of moving objects. In physics, momentum is the measure of the 'oomph' an object has due to its motion. The momentum of an object can be calculated with the equation:
\( \text{Momentum} = \text{mass} \times \text{velocity} \).

In our exercise, the twins on ice skates represent individual masses, and the action of Twin A throwing the backpack to Twin B involves changing velocities. Before the throw, the system's momentum is zero because neither twin is moving. However, as soon as Twin A throws the backpack, we calculate the backpack’s momentum using its mass and the velocity at which it was thrown. Similarly, the momentum of Twin A can be determined by considering the velocity she acquires in the opposite direction post-throw. Such calculations allow us to predict future movements based on current dynamics, which is essential for solving physics problems involving motion.

Closed System Physics

A closed system in physics is one where no mass enters or leaves the system, and no external forces act on the system. This leads to some very interesting and powerful principles, like the conservation of momentum, that help us understand how objects will behave within such a system.

In the scenario with the skating twins, the frozen lake can be considered a closed system since we're ignoring external factors like air resistance and friction from the ice. That means the initial total momentum of the system (the twins and the backpack) must be the same as the total momentum after any internal interactions take place, such as throwing and catching the backpack. Grasping the concept of a closed system is integral to predicting outcomes in isolated environments.

Law of Conservation of Momentum

The law of conservation of momentum states that within a closed system, the total momentum remains constant unless acted upon by an external force. In our exercise, this law is perfectly demonstrated: before the backpack was thrown, the total momentum of the twins and the backpack was zero, and after the throw, it had to remain zero.

This is why after Twin A throws the backpack with a certain momentum, she recoils with an equal and opposite momentum to maintain the total momentum at zero. When Twin B catches the backpack, the system still remains with zero total momentum but now redistributed differently between Twin A and Twin B plus the backpack. Understanding the conservation of momentum helps us to predict that Twin A will move in one direction and Twin B in the opposite, their individual momenta perfectly balancing out.