Problem 23 - College Physics

Question

\(\bullet\) An astronaut in space cannot use a scale or balance to weigh objects because there is no gravity. But she does have devices to measure distance and time accurately. She knows her own mass is \(78.4 \mathrm{kg},\) but she is unsure of the mass of a large gas canister in the airless rocket. When this canister is approaching her at \(3.50 \mathrm{m} / \mathrm{s},\) she pushes against it, which slows it down to 1.20 \(\mathrm{m} / \mathrm{s}\) (but does not reverse it) and gives her a speed of 2.40 \(\mathrm{m} / \mathrm{s}\) . (a) What is the mass of this canister? (b) How much kinetic energy is "lost" in this collision, and what happens to that energy?

Step-by-Step Solution

Answer

The mass of the canister is 81.8 kg. The kinetic energy lost is 217.507 J, converted to internal energy.

Step 1: Analyzing the Momentum Conservation

In an isolated system, without external forces, the total momentum before and after an interaction is conserved. We can state this principle formally as: \( m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \), where \( u \) stands for initial velocities and \( v \) for final velocities. Let's assign:\( m_1 = 78.4 \, \text{kg} \) (astronaut's mass), \( u_1 = 0 \, \text{m/s} \), \( v_1 = 2.40 \, \text{m/s} \) (astronaut's final speed), \( m_2 \) as the mass of the canister, \( u_2 = 3.50 \, \text{m/s} \) and \( v_2 = 1.20 \, \text{m/s} \).

Step 2: Setting up the Equation

Substitute the known values into the momentum conservation equation: \( 78.4 \cdot 0 + m_2 \cdot 3.50 = 78.4 \cdot 2.40 + m_2 \cdot 1.20 \). Simplify this to find: \( 3.50 m_2 = 188.16 + 1.20 m_2 \).

Step 3: Solving for the Mass of the Canister

Re-arrange the equation to solve for \( m_2 \): \[ 3.50m_2 - 1.20m_2 = 188.16 \] to \[ 2.30m_2 = 188.16 \]. Finally, solve for \( m_2 \) by dividing both sides by 2.30: \[ m_2 = \frac{188.16}{2.30} = 81.8 \, \text{kg} \].

Step 4: Finding the Kinetic Energy Before and After the Collision

Calculate the initial and final kinetic energies using \( KE = \frac{1}{2} m v^2 \). The initial kinetic energy \( KE_{initial} = \frac{1}{2} m_2 u_2^2 \) yields \( = \frac{1}{2} \times 81.8 \times (3.50)^2 \). For the astronaut after acquiring a velocity: \( = \frac{1}{2} \, \times 78.4 \, \times 0^2 = 0 \).

Step 5: Calculating Initial Kinetic Energy

Compute initial kinetic energy for the canister: \( KE_{initial} = \frac{1}{2} \, \times 81.8 \, \times (3.50)^2 = 502.195 \, \text{Joules} \).

Step 6: Finding the Final Kinetic Energy

Calculate the final kinetic energy for both objects. Canister: \( \frac{1}{2} \, \times 81.8 \, \times (1.20)^2 = 58.896 \, \text{Joules} \); Astronaut: \( \frac{1}{2} \, \times 78.4 \, \times (2.40)^2 = 225.792 \, \text{Joules} \). Total \( KE_{final} = 58.896 + 225.792 = 284.688 \).

Step 7: Calculating the Lost Kinetic Energy

Determine the kinetic energy 'lost' by taking the difference between initial and final values: \( KE_{lost} = 502.195 - 284.688 = 217.507 \, \text{Joules} \). This lost energy is converted to internal energy due to work done against forces like friction if present, or manifest as changes in internal temperature or sound in specific contexts.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is an important concept in physics, especially when analyzing motion and interactions like collisions. The formula to calculate kinetic energy is: \[ KE = \frac{1}{2} mv^2 \]where:

• \( KE \) stands for kinetic energy,
• \( m \) is the mass of the object in kilograms,
• \( v \) is the velocity of the object in meters per second.

When analyzing collisions in physics like in our astronaut problem, it's essential to determine how kinetic energy changes. During collisions, kinetic energy may not necessarily be conserved due to energy transformations, like changes into thermal energy or sound. This is because it's only conserved in perfectly elastic collisions, which rarely occur in practice. Knowing initial and final kinetic energies helps to assess any energy lost during the collision as seen when the astronaut and canister interacted in space. Kinetic energy before and after the collision in the astronaut scenario is indicative of how energy is redistributed in the system.

Astronaut Mass

Astronaut mass plays a crucial role in understanding physics problems set in space, where traditional weighing methods don't apply due to lack of gravity. Instead, one considers mass as an inherent property of the astronaut, irrespective of gravitational effects. In the original problem, the astronaut's mass is known to be 78.4 kg. This known quantity allows us to calculate the mass of the unknown item, here a gas canister, using principles like momentum conservation. When an astronaut interacts physically with an object, such as pushing a gas canister, their mass helps in calculating resulting velocities and interactions since it doesn't change regardless of environment or motion conditions. This constant mass concept is vital in space physics, as it enables proper momentum calculations and helps in determining kinetic energies before and after interactions.

Collisions in Physics

Collisions are significant events studied in physics, representing interactions where momentum and energy are transferred between objects. They are analyzed using laws of conservation, primarily momentum and sometimes energy, to understand outcomes of such interactions. Momentum conservation states that the total momentum of a closed system remains constant if no external forces act upon it. Thus, for the astronaut and canister:\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]In this formula:

• \( m_1 \) and \( m_2 \) are the masses of the astronaut and canister,
• \( u_1 \) and \( u_2 \) are their initial velocities,
• \( v_1 \) and \( v_2 \) are their final velocities.

In our scenario, this equation helped determine the mass of the canister after the collision. Furthermore, energy changes post-collision highlight conversion into other forms, albeit energy may not all be mechanical due to real-world conditions like space friction or internal changes, which dissipate some of it.

Physics Problem Solving

Physics problem solving involves structured approaches to determine unknowns in given scenarios, oftentimes involving multiple principles and formulas. The astronaut problem utilized both momentum conservation and kinetic energy calculations to solve for unknowns in the context of a collision. Step-by-step methods in solving physics problems ensure clarity and accuracy:

• First, identify known and unknown variables, using units that align with formulas, assuming correct units simplify problem analysis.
• Apply relevant physical laws, such as conservation of momentum and energy transformations.
• Calculate using mathematical formulas and rearrange them when necessary to isolate unknown variables efficiently.
• Consider real-world context, acknowledging how theoretical models may simplify or not account for certain practical factors like energy dispersal.

Developing these problem-solving skills enables one to handle complex physics scenarios, using logical reasoning and mathematical operations to derive solutions accurately.