Question

A space vehicle is traveling at \(3860 \mathrm{~km} / \mathrm{h}\) with respect to the Earth when the exhausted rocket motor is disengaged and sent backward with a speed of \(125 \mathrm{~km} / \mathrm{h}\) with respect to the command module. The mass of the motor is four times the mass of the module. What is the speed of the command module after the separation?

Short answer

The speed of the command module after the separation can be calculated using the principle of conservation of momentum. Substitute the known values into the equation from step 4 to find the result.

Step-by-step solution

Understanding the problem

We know that the initial speed of the space vehicle (which includes both the rocket and command module) is 3860 km/hr. The mass of the rocket is four times the mass of the command module. The rocket motor is disengaged and is sent backward with a speed of 125 km/hr with respect to the command module.

Define and Assign Variables

Let's denote the mass of the command module as \(m\), hence the rocket's mass is \(4m\). The initial velocity of the system \(V_{i} = 3860\ km/hr\), the rocket's speed after disconnection \(V_{r} = -125\ km/hr\) (negative indicates opposite direction) and we need to find the speed of the command module \(V_{cm}\) after separation.

Conservation of Momentum

According to the principle of conservation of momentum, the total momentum before separation equals the total momentum after separation: \[(m+4m) \cdot V_{i} = m \cdot V_{cm} + 4m \cdot V_{r}\]

Solve for the command module speed

Rearrange the equation in step 3 to solve for \(V_{cm}\): \[V_{cm} = \frac{(m+4m) \cdot V_{i} - 4m \cdot V_{r}}{m}\] Substitute the known values into the equation we get the speed of the command module after separation.

Key concepts

Space Vehicle Dynamics

Space vehicle dynamics involves understanding how different forces and motions affect the movement and control of spacecraft. This includes the different components, such as rocket motors and command modules, and how they interact during maneuvers.

In the context of the exercise, we are dealing with a space vehicle consisting of a rocket motor and a command module. When the rocket motor disengages from the module, each component follows new trajectories based on physical principles like conservation of momentum.

• The space vehicle's initial speed is crucial, showing how fast it moves in space relative to Earth. In this scenario, that's 3860 km/h.
• The dynamics change significantly once the motor disconnects and moves in the opposite direction at 125 km/h.
• Understanding these changes helps in predicting the new speed of the command module.

Every maneuver in space is delicate and must adhere to dynamics laws, ensuring successful operation and safety of the spacecraft and its crew.

Kinematics

Kinematics is the study of motion without considering forces that cause it. It looks at changes in position, velocity, and acceleration over time. In this exercise, the emphasis is on how fast the space vehicle and its components move relative to each other and the Earth.

Initially, the vehicle travels in one direction. After disengagement, parts of it move differently:

• The space vehicle's velocity before separation is crucial at 3860 km/h.
• The rocket motor's velocity changes to -125 km/h in the opposite direction once it's released. This change in direction is vital for understanding overall motion.

To find out how the command module behaves after its split from the rocket motor, we need to analyze these velocities and how they relate. Kinematic equations help solve these kinds of motion problems by providing a quantitative framework.

Mass and Velocity Relationship

The relationship between mass and velocity is fundamental to comprehending momentum changes in systems, such as when parts of a space vehicle separate. Understanding how mass and velocity interact allows us to predict changes in motion and velocity post-separation.

The principle used here is momentum conservation. It's critical in this context:

• Momentum is the product of mass and velocity: \(p = mv\).
• Before separation, the total momentum involves both the motor and command module together.
• The mass of the rocket motor is four times that of the command module, and this mass ratio heavily influences post-separation velocities.
• By conservation of momentum: the total initial momentum equals the sum of individual momenta after separation.

This relationship provides the foundation for computing the new speed of the command module once the motor disengages, through rearranging the momentum equation.