Question

An astronaut in space cannot use a conventional means, such as a scale or balance, to determine the mass of an object. But she does have devices to measure distance and time accurately. She knows her own mass is 78.4 kg, but she is unsure of the mass of a large gas canister in the airless rocket. When this canister is approaching her at 3.50 m/s, she pushes against it, which slows it down to 1.20 m/s (but does not reverse it) and gives her a speed of 2.40 m/s. What is the mass of this canister?

Short answer

The mass of the canister is approximately 81.8 kg.

Step-by-step solution

Understand the Law of Conservation of Momentum

The law of conservation of momentum states that if no external forces act on a system, the total momentum before an interaction is equal to the total momentum after. In mathematical terms, this can be written as: \ \[m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}\] \ where \(m_1\) and \(m_2\) are the masses of the astronaut and the canister, respectively, and \(v_{1i}\), \(v_{2i}\), \(v_{1f}\), and \(v_{2f}\) are their initial and final velocities.

Identify the Initial and Final Velocities

Given: - The initial velocity of the canister \(v_{2i} = 3.50\, \text{m/s}\) - The initial velocity of the astronaut \(v_{1i} = 0\, \text{m/s}\) (since she is initially at rest) - The final velocity of the canister \(v_{2f} = 1.20\, \text{m/s}\) - The final velocity of the astronaut \(v_{1f} = 2.40\, \text{m/s}\).

Write the Momentum Equation

Substitute the given values into the conservation of momentum equation: \[78.4 \times 0 + m_2 \times 3.50 = 78.4 \times 2.40 + m_2 \times 1.20\] \ This simplifies to: \[3.50m_2 = 188.16 + 1.20m_2\].

Solve for the Unknown Mass

Rearrange the equation for \(m_2\): \[3.50m_2 - 1.20m_2 = 188.16\] Simplify to get: \[2.30m_2 = 188.16\] Thus, \[m_2 = \frac{188.16}{2.30}\] Calculate \(m_2\): \[m_2 \approx 81.8 \, \text{kg}\].

Key concepts

Astronaut Physics

In the fascinating realm of astronaut physics, motion and interaction in space present unique challenges and scenarios. Unlike on Earth, astronauts operate in a microgravity environment, where conventional methods of measuring and manipulating mass do not apply. The absence of gravity means no scales or balances can be used.

To comprehend how astronauts solve such issues, consider the example of an astronaut interacting with a gas canister in the void of space. Here, the principles of momentum come into play significantly. An astronaut can use their understanding of motion to indirectly gauge the mass of objects. By measuring velocity changes and utilizing the conservation of momentum, astronauts can determine the mass of an object without relying on traditional weighing devices. These principles form a crucial part of their training and operations in space.

Velocity Calculations

Velocity calculations are fundamental to solving problems associated with the movement of objects in space. Velocity, by definition, is the speed of something in a given direction. When dealing with two objects such as an astronaut and a gas canister, understanding their respective velocities before and after an interaction helps us analyze their motion.

In the given exercise, initially, the gas canister was moving towards the astronaut at 3.50 m/s, while the astronaut was at rest with a velocity of 0 m/s. After the astronaut pushed the canister, its velocity decreased to 1.20 m/s, while the astronaut gained a velocity of 2.40 m/s in the opposite direction. These changes are pivotal in solving for unknown variables, as they reflect the action-reaction nature of the momentum exchange between the two bodies. Correctly identifying and substituting these velocities into the momentum equation allows us to deduce other quantities such as mass.

Momentum Equation

The momentum equation plays a central role in understanding and solving the problem at hand. The law of conservation of momentum tells us that the total momentum before an interaction equals the total momentum after, provided no external forces act on the objects involved.

For this specific exercise, we start with the equation:

• Initial total momentum: \[m_1v_{1i} + m_2v_{2i}\], where each term refers to the mass and initial velocity of the astronaut and canister, respectively.
• Final total momentum: \[m_1v_{1f} + m_2v_{2f}\], reflecting their masses and final velocities after interaction.

By substituting known values into this formula, you can rearrange and solve for the unknown mass of the canister. The solution requires some simple algebra, isolating the variable \(m_2\), and then performing the division to find the result. Utilizing this equation showcases the beauty of physics in which theoretical principles can provide practical solutions in the space environment.