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Chapter 9: Problem 78

Consider a rocket that is in deep space and at rest relative to an inertial reference frame. The rocket's engine is to be fired for a certain interval. What must be the rocket's mass ratio (ratio of ini. tial to final mass) over that interval if the rocket's original speed relative to the inertial frame is to be equal to (a) the exhaust speed (speed of the exhaust products relative to the rocket) and (b) \(2.0\) times the exhaust speed?

Short Answer

Expert verified
(a) Mass ratio is \( e \approx 2.718 \). (b) Mass ratio is \( e^2 \approx 7.389 \).

Step by step solution

01

Understand the Rocket Equation

In rocket propulsion, the final velocity of the rocket (\( v_f \)) can be derived using the Tsiolkovsky rocket equation:\[ v_f = v_e \cdot \ln \left( \frac{m_i}{m_f} \right) \]where \( v_e \) is the exhaust velocity, \( m_i \) is the initial mass, and \( m_f \) is the final mass of the rocket.
02

Set Up the Equation for (a)

For part (a), the final speed of the rocket \( v_f \) is equal to the exhaust speed \( v_e \). Set \( v_f = v_e \) and substitute into the rocket equation:\[ v_e = v_e \cdot \ln \left( \frac{m_i}{m_f} \right) \]This simplifies to:\[ 1 = \ln \left( \frac{m_i}{m_f} \right) \]
03

Solve the Equation for (a)

Solve the equation \( 1 = \ln \left( \frac{m_i}{m_f} \right) \) by raising \( e \) to the power of both sides:\[ e^1 = \frac{m_i}{m_f} \]Thus, the mass ratio is:\[ \frac{m_i}{m_f} = e \approx 2.718 \]
04

Set Up the Equation for (b)

For part (b), the final speed of the rocket \( v_f \) is \( 2.0 \times v_e \). Set \( v_f = 2.0 \times v_e \) and substitute into the rocket equation:\[ 2.0 \times v_e = v_e \cdot \ln \left( \frac{m_i}{m_f} \right) \]This simplifies to:\[ 2.0 = \ln \left( \frac{m_i}{m_f} \right) \]
05

Solve the Equation for (b)

Solve the equation \( 2.0 = \ln \left( \frac{m_i}{m_f} \right) \) by raising \( e \) to the power of both sides:\[ e^2 = \frac{m_i}{m_f} \]Thus, the mass ratio is:\[ \frac{m_i}{m_f} = e^2 \approx 7.389 \]

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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 an exciting and complex field that involves propelling spacecraft by expelling mass, often in the form of hot gases, in one direction to move in the opposite one. This fundamental concept stems from Newton's Third Law of Motion: for every action, there's an equal and opposite reaction.

Here鈥檚 a simple breakdown of the process:
  • Fuel burns, creating gas.
  • The gas expands and is expelled at high speed through a nozzle.
  • This expulsion pushes the rocket forward.
This propulsion technique is crucial for space travel, as it's not dependent on external media (e.g., air or water) to operate. Instead, the rocket carries all of its fuel and reacts mass (exhaust) within itself, which is especially useful in the vacuum of space.
Exhaust Velocity
Exhaust velocity plays a vital role in determining how efficiently a rocket can accelerate. It refers to the speed at which the expelled gases leave the rocket nozzle. Higher exhaust velocities result in greater thrust for the same amount of fuel.

The exhaust velocity (\(v_e\)) is determined by several factors:
  • The type of propellant used.
  • Combustion chamber pressure.
  • Nozzle design.
By optimizing these factors, engineers strive to achieve the highest possible exhaust velocity, leading to improved rocket performance. A better exhaust velocity means a more efficient transfer of energy from the fuel to the rocket鈥檚 movement.
Mass Ratio
The mass ratio is a crucial concept in understanding rocket efficiency and performance. Simply put, it's the ratio of the rocket's initial mass (\(m_i\)) to its final mass (\(m_f\)). This ratio indicates how much of the rocket's mass is used as propellant.

The Tsiolkovsky rocket equation shows that a higher mass ratio allows the rocket to achieve higher velocities. However, this also means that a larger proportion of the rocket is made of consumable fuel, which imposes design constraints:
  • Fuel constitutes a significant portion of launch weight.
  • Designs must balance fuel mass against payload and structural support.
Understanding mass ratio is key in the planning of long-duration space missions, where fuel efficiency is paramount.
Inertial Reference Frame
An inertial reference frame is a key concept in physics, providing a point of view where the laws of motion, such as Newton's laws, hold true. In the case of space travel, this is typically a frame of reference in which an object not subjected to forces continues to travel in a straight line at constant speed.

Why is this important?
  • It allows for consistent and clear calculations of motion.
  • In an inertial reference frame, you can predict how a rocket will behave when forces act upon it.
  • This frame is crucial for understanding the velocity changes during rocket propulsion.
When working with rockets, engineers and scientists consider the rocket's speed and direction relative to an inertial reference frame to ensure accurate trajectory predictions.

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Most popular questions from this chapter

A \(4.0 \mathrm{~kg}\) mess kit sliding on a frictionless surface explodes into two \(2.0 \mathrm{~kg}\) parts: \(3.0 \mathrm{~m} / \mathrm{s}\), due north, and \(5.0 \mathrm{~m} / \mathrm{s}, 30^{\circ}\) north of east. What is the original speed of the mess kit?

A \(5.0 \mathrm{~kg}\) block with a speed of \(3.0 \mathrm{~m} / \mathrm{s}\) collides with a 10 kg block that has a speed of \(2.0 \mathrm{~m} / \mathrm{s}\) in the same direction. After the collision, the \(10 \mathrm{~kg}\) block travels in the original direction with a speed of \(2.5 \mathrm{~m} / \mathrm{s}\). (a) What is the velocity of the \(5.0 \mathrm{~kg}\) block immediately after the collision? (b) By how much does the total kinetic energy of the system of two blocks change because of the collision? (c) Suppose, instead, that the \(10 \mathrm{~kg}\) block ends up with a speed of \(4.0 \mathrm{~m} / \mathrm{s}\). What then is the change in the total kinetic energy? (d) Account for the result you obtained in (c).

Two particles \(P\) and \(Q\) are released from rest \(1.0 \mathrm{~m}\) apart. \(P\) has a mass of \(0.10 \mathrm{~kg}\), and \(Q\) a mass of \(0.30 \mathrm{~kg} . P\) and \(Q\) attract each other with a constant force of \(1.0 \times 10^{-2} \mathrm{~N}\). No external forces act on the system. (a) What is the speed of the center of mass of \(P\) and \(Q\) when the separation is \(0.50 \mathrm{~m}\) ? (b) At what distance from \(P\) 's original position do the particles collide?

A ball having a mass of \(150 \mathrm{~g}\) strikes a wall with a speed of \(5.2 \mathrm{~m} / \mathrm{s}\) and rebounds with only \(50 \%\) of its initial kinetic energy. (a) What is the speed of the ball immediately after rebounding? (b) What is the magnitude of the impulse on the wall from the ball? (c) If the ball is in contact with the wall for \(7.6 \mathrm{~ms}\), what is the magnitude of the average force on the ball from the wall during this time interval?

A projectile proton with a speed of \(500 \mathrm{~m} / \mathrm{s}\) collides elastically with a target proton initially at rest. The two protons then move along perpendicular paths, with the projectile 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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