Problem 37 - College Physics

Question

A \(60-\mathrm{kg}\) astronaut floating at rest in space outside a space capsule throws his \(0.50-\mathrm{kg}\) hammer such that it moves with a speed of \(10 \mathrm{~m} / \mathrm{s}\) relative to the capsule. What happens to the astronaut?

Step-by-Step Solution

Answer

The astronaut moves backward at approximately \(-0.0833 \text{ m/s}\) relative to the capsule.

Step 1: Understanding Conservation of Momentum

In an isolated system, the total momentum before and after an event must remain constant. This is known as the law of conservation of momentum. For the astronaut and the hammer, which are initially at rest, the total initial momentum is zero.

Step 2: Writing the Momentum Equation

Let the velocity of the hammer be \(v_h = 10 \text{ m/s}\) and the mass \(m_h = 0.50 \text{ kg}\). The astronaut's mass is \(m_a = 60 \text{ kg}\), and we need to find the astronaut's velocity \(v_a\). Since the initial momentum is zero, the equation for the conservation of momentum is:\[0 = m_h \cdot v_h + m_a \cdot v_a\].

Step 3: Solving for the Astronaut's Velocity

Rearrange the equation to solve for \(v_a\):\[m_a \cdot v_a = - m_h \cdot v_h\]. Substitute the known values: \[(60 \text{ kg}) \cdot v_a = -(0.50 \text{ kg}) \cdot (10 \text{ m/s})\].

Step 4: Calculating the Astronaut's Velocity

Solve for \(v_a\) by performing the multiplication and division: \[v_a = -\frac{0.50 \text{ kg} \times 10 \text{ m/s}}{60 \text{ kg}}\].\[v_a = -0.0833 \text{ m/s}\]. The negative sign indicates that the astronaut moves in the opposite direction to the hammer.

Key Concepts

Momentum Equation

The momentum equation is a fundamental part of physics, specifically in dynamics. It describes how momentum, a measure of an object's motion, is calculated. Momentum (\(p\)) is defined as the product of an object's mass (\(m\)) and its velocity (\(v\)):

• \(p = m \, \cdot \, v\)

In the context of this exercise, the astronaut and the hammer initially have zero total momentum because they are at rest. According to the conservation of momentum, the total momentum of an isolated system should remain constant. This principle directs us to write the momentum equation,

• \(0 = m_h \, \cdot \, v_h + m_a \, \cdot \, v_a\)

to solve for the unknown velocity of the astronaut (\(v_a\)). This equation encompasses both the mass and velocity of the hammer and astronaut, linking the two values together before and after the hammer is thrown.

Astronaut Velocity

After the hammer is thrown, the astronaut will move in the opposite direction. This change in motion is explained by the equation derived from the conservation of momentum principles. Substituting the known values,

• the hammer's mass \(m_h = 0.50 \, \mathrm{kg}\),
• its velocity \(v_h = 10 \, \mathrm{m/s}\),
• and the astronaut's mass \(m_a = 60 \, \mathrm{kg}\),

into the momentum equation, we solve for the astronaut's velocity (\(v_a\)). The calculation is straightforward:\[m_a \, \cdot \, v_a = -m_h \, \cdot \, v_h\]\[v_a = -\frac{0.50 \, \mathrm{kg} \, \times \, 10 \, \mathrm{m/s}}{60 \, \mathrm{kg}}\]This simplifies to \(v_a = -0.0833 \, \mathrm{m/s}\). The negative sign is crucial as it indicates the direction of motion, showing that the astronaut moves in the opposite direction of the hammer.

Isolated System

For the conservation of momentum to hold true, we must consider an isolated system. In physics, an isolated system is one where no external forces act on the bodies within it. Since the astronaut and hammer are floating in space, they form an almost perfect example of an isolated system. The absence of external forces ensures that their total momentum before and after the hammer is thrown remains constant. Being free from other influences, such as gravity, air resistance, or external mechanical forces, means that the only change occurring in the system is internal (between the astronaut and hammer). Therefore, the principle of conservation of momentum can be applied confidently, allowing accurate calculation of the astronaut's velocity after the action of throwing the hammer.

Law of Conservation of Momentum

The law of conservation of momentum is a key concept in physics that states that within an isolated system, the total momentum remains constant if no external forces are acting on it. This law implies that any change in momentum of one object must be balanced by an equal and opposite change in momentum of another. In this exercise,

• the total initial momentum of the astronaut and hammer system is zero,
• and despite the hammer acquiring momentum when thrown, the system's overall momentum does not change, remaining zero.

The hammer's gained momentum is exactly countered by the momentum of the astronaut moving in the opposite direction. This balancing act under the law of conservation of momentum is not only pivotal in this scenario but serves as a foundational principle in many other physics problems. It emphasizes the interdependence of objects in a system and ensures that in the absence of external influences, motion is always conserved.