Problem 51 - College Physics

Question

A \(70 \mathrm{~kg}\) astronaut floating in space in a \(110 \mathrm{~kg}\) MMU (manned maneuvering unit) experiences an acceleration of \(0.029 \mathrm{~m} / \mathrm{s}^{2}\) when he fires one of the MMU's thrusters. (a) If the speed of the escaping \(\mathrm{N}_{2}\) gas relative to the astronaut is \(490 \mathrm{~m} / \mathrm{s},\) how much gas is used by the thruster in \(5.0 \mathrm{~s} ?\) (b) What is the thrust of the thruster?

Step-by-Step Solution

Answer

(a) 0.0532 kg of gas; (b) Thrust is 5.22 N.

Step 1: Determine Total Mass

Calculate the total mass of the astronaut and the MMU. The astronaut weighs 70 kg and the MMU weighs 110 kg, so the total mass is:\[ m = 70 \, \text{kg} + 110 \, \text{kg} = 180 \, \text{kg} \]

Step 2: Calculate Force from Acceleration

Use Newton's second law to calculate the force (thrust) produced by the thruster. The force can be calculated as:\[ F = m \times a = 180 \, \text{kg} \times 0.029 \, \text{m/s}^2 = 5.22 \, \text{N} \]

Step 3: Calculate Mass of Gas Used

To find the mass of gas expelled, use the momentum principle. The change in momentum over 5 seconds equals the force times the time:\[ F \times \Delta t = \Delta p = v \times \Delta m \] where \(v\) is the speed of the escaping gas and \(\Delta m\) is the mass of the gas. Solving for \(\Delta m\):\[ \Delta m = \frac{F \times \Delta t}{v} = \frac{5.22 \, \text{N} \times 5.0 \, \text{s}}{490 \, \text{m/s}} \approx 0.0532 \, \text{kg} \]

Step 4: Confirm the Thrust

The thrust calculated in step 2 was 5.22 N. This should be the constant thrust produced by the thruster throughout the gas expulsion process.

Key Concepts

Momentum Principle

The momentum principle is at the heart of understanding how forces and motions are interrelated, especially in space physics. In essence, it states that the change in momentum of an object is equal to the impulse applied to it.\[ p = mv \]Here, \( p \) represents momentum, \( m \) is mass, and \( v \) is velocity.
When solving the exercise, we used this principle to determine the amount of gas expelled by calculating the change in momentum and equating it to the product of force and time: \( F \times \Delta t = v \times \Delta m \). This relationship is essential because it helps us see how applying a force over a duration changes the mass flow rate of the rocket's fuel, crucial for real-life applications in space travel.

• The momentum principle links force, mass, and velocity, pivotal when calculating exhaust in space propulsion systems.
• The resulting change in mass, when rearranged, shows the inverse relationship between how much gas is used and its velocity.

Thrust Calculation

Thrust is the force exerted by a rocket engine or thruster to propel a spacecraft. Calculating thrust involves using Newton's Second Law of Motion, which states that the force exerted on an object equals its mass times its acceleration (\( F = ma \)). In the exercise, we calculated the thrust generated by the MMU's thruster to be \( 5.22 \text{ N} \).
Understanding thrust calculation is crucial because:

• It determines how quickly a spacecraft can change its velocity in a given direction.
• Higher thrust means greater acceleration for a given mass.
• It informs engineers how much force a thruster needs to produce to achieve desired maneuvers.

The thrust resulting from the escaping gas creates the necessary force to accelerate both the astronaut and the MMU together.

Mass Calculation

Mass calculation helps determine the total weight being moved or affected in various scenarios. In space physics, it is crucial to understand the combined mass of the objects being acted upon. In our exercise, we summed the astronaut's and MMU's masses---70 kg and 110 kg, respectively---to obtain a total mass of 180 kg.
Calculating combined mass is vital for several reasons:

• It allows for accurate force calculations, as seen in \( F = ma \).
• Mass affects the object's momentum and how it changes under force, as per \( \Delta p = m \cdot \Delta v \).
• Understanding mass helps ensure precise fuel and thrust requirements, pivotal for effective space missions.

It underscores the importance of considering all contributing factors to the total mass to achieve desired motion effects.

Space Physics

Space physics explores how physical laws operate within the context of space. This field especially focuses on understanding how momentum, force, and mass interrelate in an environment void of atmospheric resistance. The exercise illustrates basic principles of space physics through the astronaut and MMU scenario, highlighting the fundamentals like if an external force acts on the system, it directly impacts acceleration.
Key considerations in space physics include:

• The absence of friction and air resistance emphasizes the need for precise thrust and mass calculations.
• Rockets and spacecraft must rely on their own propulsion systems to maneuver since no external forces like air currents exist.
• Space physics explains why even small forces can lead to significant changes in motion over time due to low resistance.

The principles of space physics ensure that astronauts can control and change their movements precisely, using thrusters effectively to navigate the vast emptiness of space.