Question

The last stage of a rocket is traveling at a speed of \(7600 \mathrm{~m} / \mathrm{s}\). This last stage is made up of two parts that are clamped together - namely, a rocket case with a mass of \(290.0 \mathrm{~kg}\) and a payload capsule with a mass of \(150.0 \mathrm{~kg}\). When the clamp is released, a compressed spring causes the two parts to separate with a relative speed of \(910.0 \mathrm{~m} / \mathrm{s}\). ( \(a\) ) What are the speeds of the two parts after they have separated? Assume that all velocities are along the same line. \((b)\) Find the total kinetic energy of the two parts before and after they separate and account for the difference, if any.

Short answer

The velocities and kinetic energies would need to be computed after setting up and solving the equations as described in the steps. The final answers will depend on the results from solving the equations.

Step-by-step solution

Calculate velocities after separation

Conservation of momentum denotes that the total momentum before separation equals to the total momentum after separation. The total momentum before separation is the sum of the momentum of the rocket case and the payload capsule, which equals to \( (290.0 \mathrm{~kg} + 150.0 \mathrm{~kg}) * 7600 \mathrm{~m/s} \), as both are traveling at the same speed. After separation, the rocket case and the payload capsule move at different speeds, let's assume the final velocity of rocket case as \(V_1\) and payload capsule as \(V_2\), so the conservation of momentum equation can be written as: \(290.0 \mathrm{~kg} * V_1 + 150.0 \mathrm{~kg} * V_2 = (290.0 \mathrm{~kg} + 150.0 \mathrm{~kg}) * 7600 \mathrm{~m/s}\). Because the relative speed of the two parts after they separate is 910 m/s, we can write equation \(V_2 - V_1 = 910 \mathrm{~m/s}\). Solving those two equations will derive \(V_1\) and \(V_2\).

Calculate total kinetic energy before and after separation

The kinetic energy (KE) of a body is given by the formula \( \frac{1}{2}mv^2 \) where m is the mass and v is velocity. Thus, the total kinetic energy before separation equals to \( \frac{1}{2}*(290.0 \mathrm{~kg} + 150.0 \mathrm{~kg})*7600^2 \mathrm{~m^2/s^2}\). After separation, the total kinetic energy is the sum of the kinetic energies of the rocket case and the payload capsule which equals to \( \frac{1}{2}*290.0 \mathrm{~kg}*V_1^2 + \frac{1}{2}*150.0 \mathrm{~kg}*V_2^2 \). Finding the difference between the total kinetic energy before and after the separation will show whether there has been an increase or decrease in kinetic energy after the separation.

Key concepts

Kinetic Energy

Kinetic energy is the energy a body possesses due to its motion. It's an important concept in physics, especially when effects like speed and mass are involved in actions such as rocket separations. The formula for kinetic energy is given by \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass, and \( v \) is the velocity of the object.
The kinetic energy of a system gives us insights into how energy is distributed between objects during movement.

• Before separation, the total kinetic energy was calculated using the combined mass and speed of the rocket's last stage.
• After separation, each part of the rocket (the rocket case and the payload) has its own kinetic energy.

By comparing the kinetic energy before and after separation, we may find "lost" or "gained" energy. This helps us understand energy transformations during the separation process.
In the given exercise, solving for kinetic energy helps to determine how the separation affects overall energy distribution, showing the practical applications of theoretical physics concepts.

Rocket Separation

Rocket separation is a critical phase where different parts of a rocket are decoupled from each other, typically involving mechanisms like compressed springs. This phase is essential for space missions where different stages of the rocket complete their intended functions and separate to allow subsequent stages to carry on with the mission.
During the process of rocket separation, principles like the conservation of momentum are crucial. This principle ensures that the momentum of the system remains constant before and after separation.

• Each part moves away at different velocities, calculated based on mass and the relative speed post-separation.
• The separation must be carefully managed to ensure safety and mission success.

In our example, a compressed spring creates the force needed to separate the rocket case from the payload capsule. Understanding these dynamics proves vital to engineering safe and efficient separation sequences in rocket science, highlighting why it's a frequently studied process.

Relative Velocity

Relative velocity serves as a way to measure how fast one object moves in comparison to another. It's a fundamental concept for understanding objects in motion, especially in mechanics and aerodynamics, like when calculating separation phases in rockets.
When two parts of a body, such as a rocket, separate, they do so with a specific relative velocity. This means one part is moving faster than the other by a certain speed.

• This relative velocity of 910 m/s indicates how quickly the payload is separating from the rocket case.
• Using both relative velocity and conservation of momentum, the final speeds of the separated parts can be derived.

In practical terms, relative velocity is crucial for predicting and controlling behavior during separation, aiding in the precision needed for planned trajectories and ensuring that operations proceed as intended. In the given problem, this relative velocity helps solve for the rockets' speed post-separation, forming a vital part of motion analysis.