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Chapter 9: Problem 127

During a lunar mission, it is necessary to increase the speed of a spacecraft by \(2.2 \mathrm{~m} / \mathrm{s}\) when it is moving at \(400 \mathrm{~m} / \mathrm{s}\) relative to the Moon. The speed of the exhaust products from the rocket cngine is \(1000 \mathrm{~m} / \mathrm{s}\) relative to the spacecraft. What fraction of the initial mass of the spacecraft must be burned and cjected to accomplish the speed increase?

Short Answer

Expert verified
The fraction of the initial mass to be burned is approximately 0.0022.

Step by step solution

01

Identify the Known Variables

The current speed of the spacecraft is \( v_i = 400 \, \text{m/s} \). The required increase in speed is \( \Delta v = 2.2 \, \text{m/s} \). The speed of the exhaust relative to the spacecraft is \( v_e = 1000 \, \text{m/s} \). We need to find the fraction of the initial mass that must be burned to increase the speed.
02

Apply the Rocket Equation

We use the Tsiolkovsky rocket equation: \( \Delta v = v_e \cdot \ln \left(\frac{m_i}{m_f}\right) \), where \( m_i \) is the initial mass and \( m_f \) is the final mass of the spacecraft. Solve this for the mass ratio \( \frac{m_i}{m_f} \).
03

Solve for Mass Ratio

Rearrange the rocket equation to find \( \ln \left(\frac{m_i}{m_f}\right) = \frac{\Delta v}{v_e} = \frac{2.2}{1000} = 0.0022 \). Therefore, \( \frac{m_i}{m_f} = e^{0.0022} \approx 1.0022 \).
04

Calculate Mass Fraction Ejected

The mass fraction ejected \( f \) is given by \( f = 1 - \frac{m_f}{m_i} = 1 - \frac{1}{1.0022} \approx 0.0022 \). This is the fraction of the initial mass that must be burned and ejected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rocket Propulsion
Rocket propulsion is the process that allows spacecraft to move from one place to another in space. It plays a crucial role in enabling missions like lunar landings or interplanetary voyages. The basic principle behind rocket propulsion is Newton's third law of motion: for every action, there is an equal and opposite reaction.
Imagine throwing a ball while standing on a skateboard. The force of throwing pushes the ball forward, while the skateboard moves backward in response. Similarly, rockets expel mass in one direction (exhaust gases) and move in the opposite direction.
  • Rockets require fuel, which when burned, produces high-speed exhaust gases.
  • These gases are expelled at high velocity, creating thrust.
  • The direction and amount of thrust determine the rocket's movement.
In the given exercise, the rocket propulsion system helps the spacecraft gain additional speed, vital for maneuvering toward the Moon.
Mass Ejection
Mass ejection is a fundamental aspect of how rockets generate the force needed for propulsion. It involves the process of expelling part of the spacecraft's mass, usually in the form of exhaust gases, to achieve thrust.
This concept ties back to the Tsiolkovsky rocket equation, which calculates the necessary change in velocity (\(\Delta v\)) by considering the speed and mass of the exhaust.
To elaborate further:
  • The mass that is initially part of the spacecraft is later transformed into high-speed exhaust gases.
  • This mass, when ejected, generates momentum needed to change the spacecraft's speed.
  • The relationship between the velocity of the ejected mass and the required change in speed is mathematically explained by the rocket equation.
Through efficient mass ejection, the spacecraft accomplishes the desired speed increase noted in the exercise: from 400 m/s to 402.2 m/s relative to the Moon.
Orbital Mechanics
Orbital mechanics is the field of study that explores the motion of objects in space under the influence of gravitational forces. It's essential for understanding how spacecraft move, reach their destinations, and perform orbital maneuvers.
In the context of the exercise, the spacecraft's motion around the Moon is a practical application of orbital mechanics principles.
  • Gravity influences every motion; it's the force keeping an orbiting object on its path.
  • By changing a spacecraft鈥檚 velocity, you alter its orbit. A small change in speed (such as 2.2 m/s) can result in significant orbital shifts.
  • The calculated ejection of mass is carefully designed to perform such orbital adjustments efficiently.
Understanding these principles allows mission planners to execute precise maneuvers as they navigate the complexities of space travel.

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Most popular questions from this chapter

A steel ball of mass \(0.500 \mathrm{~kg}\) is fastened to a cord that is \(70.0 \mathrm{~cm}\) long and fixed at the far end. The ball is then released when the cord is horizontal (Fig. \(9-65\) ). At the bottom of its path, the ball strikes a \(2.50 \mathrm{~kg}\) steel block initially at rest on a frictionless surface. The collision is elastic. Find (a) the speed of the ball and (b) the speed of the block, both just after the collision.

particle 1 of mass \(m_{1}=0.30 \mathrm{~kg}\) slides rightward along an \(x\) axis on a frictionless floor with a speed of \(2.0 \mathrm{~m} / \mathrm{s}\). When it reaches \(x=0,\) it undergocs a one-dimensional elastic collision with stationary particle 2 of mass \(m_{2}=0.40 \mathrm{~kg}\). When particle 2 then reaches a wall at \(x_{n}=70 \mathrm{~cm},\) it bounces from the wall with no loss of speed. At what position on the \(x\) axis does particle 2 then collide with particle \(1 ?\)

Particle \(A\) and particle \(B\) are held together with a com- pressed spring between them. When they are released, the spring pushes them apart, and they then fly off in opposite directions, free of the spring. The mass of \(A\) is 2.00 times the mass of \(B,\) and the energy stored in the spring was 60 J. Assume that the spring has negligible mass and that all its stored energy is transferred to the particles. Once that transfer is complete, what are the kinetic energics of (a) particle \(A\) and (b) particle \(B ?\)

A \(6090 \mathrm{~kg}\) space probe moving nose-first toward Jupiter at \(105 \mathrm{~m} / \mathrm{s}\) relative to the Sun fires its rocket engine, ejecting \(80.0 \mathrm{~kg}\) of exhaust at a speed of \(253 \mathrm{~m} / \mathrm{s}\) relative to the space probe. What is the final velocity of the probe?

A projectile proton with a speed of \(500 \mathrm{~m} / \mathrm{s}\) collides elas- tically with a target proton initially at rest. The two protons then move along perpendicular paths, with the projcctile path at \(60^{\circ}\) from the original direction. After the collision, what are the speeds of (a) the target proton and (b) the projectile proton?
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