Question

A 70-kg astronaut floating in space in a 110-kg MMU (manned maneuvering unit) experiences an acceleration of 0.029 m/s\(^2\) when he fires one of the MMU's thrusters. (a) If the speed of the escaping N\(_2\) gas relative to the astronaut is 490 m/s, how much gas is used by the thruster in 5.0 s? (b) What is the thrust of the thruster?

Short answer

(a) 0.0533 kg of gas. (b) Thrust is 5.22 N.

Step-by-step solution

Determine the Total Mass

To find the total mass affected by the thruster's force, add the mass of the astronaut to the mass of the MMU. \[m = 70 \text{ kg (astronaut)} + 110 \text{ kg (MMU)} = 180 \text{ kg}\]So, the total mass is 180 kg.

Calculate the Thrust (Force) Produced by the Thruster

Using Newton's second law, we calculate the thrust force, \( F \), exerted when the acceleration, \( a \), is given:\[F = ma = 180 \text{ kg} \times 0.029 \text{ m/s}^2 = 5.22 \text{ N}\]Thus, the thrust of the thruster is 5.22 N.

Find the Mass Flow Rate of Escaping Gas

The thrust (force) can be related to the velocity of the escaping gas \( v \) and the mass flow rate \( \dot{m} \) by the formula:\[F = \dot{m} \times v\]Rearrange to solve for \( \dot{m} \) (mass flow rate):\[\dot{m} = \frac{F}{v} = \frac{5.22 \text{ N}}{490 \text{ m/s}}\]\[\dot{m} \approx 0.01065 \text{ kg/s}\]So, the mass flow rate of the gas is approximately 0.01065 kg/s.

Calculate the Total Mass of Gas Used in 5 Seconds

To find how much gas is used in 5 seconds, multiply the mass flow rate by the time:\[m_{\text{gas}} = \dot{m} \times \text{time} = 0.01065 \text{ kg/s} \times 5 \text{ s}\]\[m_{\text{gas}} \approx 0.05325 \text{ kg}\]Thus, approximately 0.05325 kg of gas is used in 5 seconds.

Key concepts

Newton's Second Law

Newton's Second Law of Motion is a fundamental principle in physics. It tells us how the velocity of an object changes when it is subjected to an external force. According to this law, the force acting on an object is equal to the mass of the object multiplied by its acceleration. This can be expressed with the formula:
\[ F = ma \]
where:

• \( F \) is the force (in newtons, N),
• \( m \) is the mass of the object (in kilograms, kg),
• \( a \) is the acceleration (in meters per second squared, m/s²).

Newton's Second Law helps us understand how much force is needed to move an object or resist a change in its motion. In the case of the astronaut and the MMU, we combine their masses to find the total mass. Then, we use the acceleration given to calculate the thrust force, which is the force produced by the thruster.
Understanding how this law works is crucial in many applications, from calculating weaponry trajectories to designing space vehicles.

Thrust Calculation

Thrust is the force that moves an object through a medium or space. When calculating thrust, we need to consider several important variables. Thrust can be determined using the formula derived from Newton's Second Law:
\[ F = ma \]
where:

• \( F \) is the thrust or force exerted by the thruster,
• \( m \) is the total mass of the system being accelerated (in this case, the astronaut and the MMU),
• \( a \) is the acceleration provided by the thruster.

For the astronaut problem, the total mass combined with the acceleration gives us a thrust of 5.22 N. This illustrates how thrust is directly linked to how fast the astronaut can accelerate in space, providing a foundational concept for rocket science and aerospace engineering.

Mass Flow Rate

Mass flow rate is an essential concept in fluid dynamics and propulsion. It refers to the amount of mass passing through a given point in a system per unit time. For rocket engines, it is a crucial parameter, linking the thrust of an engine to the speed of the exhaust gas.
The relationship is given by the equation:
\[ F = \dot{m} \times v \]
where:

• \( F \) is thrust,
• \( \dot{m} \) is the mass flow rate (in kilograms per second, kg/s),
• \( v \) is the exhaust velocity (in meters per second, m/s).

For the astronaut's thruster, we found that the mass flow rate is approximately 0.01065 kg/s. This value tells us how quickly the thruster is consuming gas to maintain the given acceleration. This understanding is vital for designing efficient propulsion systems with the desired thrust while managing fuel consumption.

Rocket Propulsion

Rocket propulsion is the mechanism by which rockets are thrust forward into space. It is grounded in Newton's Third Law, which states that every action has an equal and opposite reaction. In simple terms, when gas is expelled out of a rocket engine at high speed, the rocket is pushed in the opposite direction. The key to rocket propulsion lies in the acceleration of propellant mass as it exits the engine. The faster the exhaust is ejected, the greater the thrust produced. This relationship is significant in understanding how adjustments in variables like mass flow rate and exhaust speed affect flight dynamics. High exhaust velocities, cojoined with optimal mass flows, offer efficient propulsion, as exemplified by the astronaut's MMU.
Understanding rocket propulsion helps bridge the gap between theoretical physics and practical engineering, guiding the design of efficient spacecraft.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause this motion. When analyzing the motion of the astronaut and the MMU, kinematics provides the language and tools we need to describe their velocities, accelerations, and displacements over time. Even in the absence of external forces, such as gravity in the vacuum of space, kinematics still allows us to predict the motion of the astronaut based on initial conditions and applied forces. In this scenario, the acceleration provided by the thruster is a crucial starting point for analyzing how the astronaut's velocity changes over time.
Kinematic equations can be further used to determine position and velocity at any given time, making them indispensable for planning and executing maneuvers in aerospace applications.