Problem 57 - College Physics

Question

\(\bullet\) 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) Approx. 0.053 kg of gas, (b) Thrust is 5.22 N.

Step 1: Determine the Total Mass of the System

To find the total mass, sum up the mass of the astronaut and the mass of the MMU:\[m_{total} = m_{astronaut} + m_{MMU} = 70 \, \text{kg} + 110 \, \text{kg} = 180 \, \text{kg}\]

Step 2: Calculate the Thrust of the Thruster

Use Newton's second law to find the force (thrust) applied by the thruster.\[F = m_{total} \times a = 180 \, \text{kg} \times 0.029 \, \text{m/s}^2 = 5.22 \, \text{N}\]This force is the thrust provided by the MMU thruster.

Step 3: Determine the Mass of Gas Used

The momentum change equals the thrust times the time, and is also equal to the change in momentum of the gas. Use the momentum conservation principle:\[F \times t = m_{gas} \times v_{gas}\]Solving for \(m_{gas}\):\[m_{gas} = \frac{F \times t}{v_{gas}}\]With the given values:\[m_{gas} = \frac{5.22 \, \text{N} \times 5.0 \, \text{s}}{490 \, \text{m/s}} \approx 0.053 \, \text{kg}\]This is the mass of the gas used by the thruster in 5 seconds.

Key Concepts

Momentum Conservation

A fundamental principle in physics, momentum conservation is pivotal in understanding how objects move. It states that the total momentum of a closed system remains constant if no external forces act on it. In the context of space mechanics, where external forces are minimal, this becomes essential. When the astronaut and the manned maneuvering unit (MMU) fire their thruster, the escaping gases provide thrust by exerting a force in one direction, propelling the astronaut in the opposite direction due to conservation of momentum. Using the equation for momentum conservation, we find that the momentum transferred to the gas equals the momentum gained by the astronaut and MMU:

• Initial Momentum = Final Momentum
• \[ m_{gas} \times v_{gas} = (m_{total}) \times \Delta v \]

In essence, as the gas escapes with a high velocity, it pushes the astronaut in the opposite direction, illustrating Newton's Third Law: for every action, there is an equal and opposite reaction.

Thrust Calculation

The calculation of thrust is critical in applications such as rocketry and space travel. Thrust is the force produced by the thruster, allowing movement or acceleration of the system. Using Newton's Second Law, we determine the thrust by multiplying the total mass of the astronaut and MMU by the acceleration experienced:

• \[ F = m_{total} \times a \]
• In our case, \( F = 180 \times 0.029 = 5.22 \text{ N} \).

This force is what allows the astronaut to maneuver, providing the necessary velocity change. Calculating the thrust helps in understanding how much force the thruster must exert and ensures that the system behaves as desired in the context of space operations.

Space Mechanics

Space mechanics deals with the motion of objects in the vacuum of space, where traditional forces such as gravity and friction are negligible. Understanding how to control and predict the path of objects is essential. When dealing with space mechanics, concepts like momentum conservation and thrust are crucial. The forces at play are extremely precise and must be carefully calculated and executed for successful missions.

Without the influence of atmospheric forces, MMUs gain importance. They offer stability and control by using thrusters to adjust their position and orientation. This machine exemplifies space mechanics, relying on principles like controlled thrust and efficient propulsion systems to move in desired directions. The astronaut's ability to control their trajectory precisely showcases space mechanics' practical application. Implementing Newton's laws in a microgravity environment shows how theoretical physics translates into real-world action, a necessity for advancing space exploration and travel.