/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 72 A small rocket of initial mass \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 72

A small rocket of initial mass \(m_{0}\) is fired vertically upward near the surface of the earth ( \(g\) constant). If air resistance is neglected, determine the manner in which the mass \(m\) of the rocket must vary as a function of the time \(t\) after launching in order that the rocket may have a constant vertical acceleration \(a\) with a constant relative velocity \(u\) of the escaping gases with respect to the nozzle.

Short Answer

Expert verified
The mass function is \( m(t) = m_0 e^{\frac{(a + g)}{u} t} \).

Step by step solution

01

Understand the Problem

We need to find how the mass of the rocket changes over time such that the rocket experiences a constant acceleration when launched vertically. The rocket's upward force changes due to the decreasing mass and the thrust from the escaping gases.
02

Write Down the Forces

The forces acting on the rocket include the gravitational force and the thrust. The force equation in this case is given by: \[ F = m(t) rac{dv}{dt} = T - m(t)g \]where \( T \) is the thrust exerted by the escaping gases, \( m(t) \) is the mass of the rocket at time \( t \), and \( g \) is the gravitational force.
03

Consider Thrust Equation

The thrust \( T \) is generated by the speed of gases \( u \) relative to the nozzle and the rate of change of mass of the rocket. Therefore, the thrust can be written as: \[ T = u \frac{dm}{dt} \]
04

Substitute for Thrust

Substitute the expression for thrust into the force equation: \[ m(t)a = u \frac{dm}{dt} - m(t)g \]where \( a \) is the constant upward acceleration desired.
05

Solve for Mass Change Rate

Rearrange the equation to isolate the term with rate of change of mass \( \frac{dm}{dt} \): \[ u \frac{dm}{dt} = m(t)(a + g) \] Thus we have the differential equation: \[ \frac{dm}{dt} = \frac{m(t)(a + g)}{u} \]
06

Integrate to Find Mass Function

Separate the variables and integrate to solve for mass:\[ \int \frac{1}{m} dm = \int \frac{(a + g)}{u} dt \]This leads to: \[ \ln m = \frac{(a + g)}{u} t + C \] where \( C \) is the integration constant.
07

Solve for Mass at Time t

Exponentiate both sides to find \( m(t) \): \[ m(t) = e^{\left(\frac{(a + g)}{u} t + C\right)} \] Simplifying, we have: \[ m(t) = m_0 e^{\frac{(a + g)}{u} t} \] where \( m_0 = e^C \), the initial mass at \( t = 0 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Rocket Dynamics
Rocket dynamics deals with the motion of rockets influenced by forces such as thrust and gravity. The key principle is that rockets move by expelling mass in the form of high-velocity gases. Rockets are primarily governed by Newton's laws of motion, and their dynamics are significantly impacted by their mass variation during flight. Understanding rocket dynamics is crucial for predicting and controlling the motion of rockets.
  • Forces acting on a rocket include thrust, which propels it upward, and gravitational pull, which pulls it downward.
  • The rocket's propulsion comes from the high-speed ejection of exhaust gases, resulting in a reactive force known as thrust.
  • As the rocket expends fuel, its mass decreases, affecting its acceleration and velocity.
This continual change in mass creates unique challenges in predicting rocket trajectories, thereby requiring precise calculations and adjustments to maintain desired flight paths.
Mass Variation
Mass variation is a defining characteristic of rocket flight. Unlike most terrestrial vehicles, a rocket's mass decreases over time as fuel burns and is expelled as exhaust gases. This loss of mass is crucial for the rocket's acceleration and requires careful calculation to ensure the rocket achieves its mission objectives.
  • The change in mass is a function of time; as fuel is used, the rocket becomes lighter.
  • This mass loss affects the rocket's inertia and the force required to maintain a specific acceleration.
  • The accurate calculation of mass variation is vital for achieving a constant acceleration despite the loss of mass.
Understanding this concept helps in solving problems like the one presented, where the challenge is to predict how mass must change for a given acceleration.
Newton's Second Law
Newton's Second Law relates to the force, mass, and acceleration of an object. It establishes that the force acting on an object is equal to the mass of that object multiplied by its acceleration, expressed as: \( F = m rac{dv}{dt} \).
  • In the context of rockets, the forces considered are thrust and gravity.
  • Since a rocket's mass changes over time, this law is applied in a way that takes the mass variation into account.
  • The equation must consider both the changing mass and the constant acceleration requirements. This implies that the thrust must adjust according to the changing mass and the gravitational force.
Applying Newton's Second Law with the thrust equation allows calculation of how thrust and mass interact to achieve constant acceleration.
Thrust Equation
The thrust equation is essential in understanding how rockets achieve lift-off and motion. It defines the relationship between the exhaust velocity, the rate of change of mass, and the thrust produced.
  • The thrust \( T \) is calculated as the product of mass flow rate \( rac{dm}{dt} \) and the exhaust velocity \( u \). This is expressed as \( T = u rac{dm}{dt} \).
  • The thrust must counteract gravitational forces and provide enough upward force to accelerate the rocket upward.
  • In problems involving constant acceleration, the thrust must adapt to changing mass values to keep the rocket accelerating properly.
This equation is vital for solving questions on how rockets maintain a steady acceleration path, balancing between changing mass and required force.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The sprinkler is made to rotate at the constant angular velocity \(\omega\) and distributes water at the volume rate \(Q .\) Each of the four nozzles has an exit area \(A\). Water is ejected from each nozzle at an angle \(\phi\) that is measured in the horizontal plane as shown. Write an expression for the torque \(M\) on the shaft of the sprinkler necessary to maintain the given motion. For a given pressure and thus flow rate \(Q\), at what speed \(\omega_{0}\) will the sprinkler operate with no applied torque? Let \(\rho\) be the density of water.

A coil of heavy flexible cable with a total length of \(100 \mathrm{m}\) and a mass of \(1.2 \mathrm{kg} / \mathrm{m}\) is to be laid along a straight horizontal line. The end is secured to a post at \(A,\) and the cable peels off the coil and emerges through the horizontal opening in the cart as shown. The cart and drum together have a mass of \(40 \mathrm{kg} .\) If the cart is moving to the right with a velocity of \(2 \mathrm{m} / \mathrm{s}\) when \(30 \mathrm{m}\) of cable remain in the drum and the tension in the rope at the post is \(2.4 \mathrm{N},\) determine the force \(P\) required to give the cart and drum an acceleration of \(0.3 \mathrm{m} / \mathrm{s}^{2},\) Neglect all friction.

The total linear momentum of a system of five particles at time \(t=2.2 \mathrm{s}\) is given by \(\mathrm{G}_{22}=3.4 \mathrm{i}-\) \(2.6 j+4.6 \mathrm{k} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} .\) At time \(t=2.4 \mathrm{s},\) the linear momentum has changed to \(\mathbf{G}_{2,4}=3.7 \mathbf{i}-2.2 \mathbf{j}+4.9 \mathbf{k}\) \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s} .\) Calculate the magnitude \(F\) of the time average of the resultant of the external forces acting on the system during the interval.

The experimental ground-effect machine has a total weight of 4200 lb. It hovers 1 or \(2 \mathrm{ft}\) off the ground by pumping air at atmospheric pressure through the circular intake duct at \(B\) and discharging it horizontally under the periphery of the skirt \(C\). For an intake velocity \(v\) of \(150 \mathrm{ft} / \mathrm{sec},\) calculate the aver age air pressure \(p\) under the 18 -ft-diameter machine at ground level. The specific weight of the air is \(0.076 \mathrm{lb} / \mathrm{ft}^{3}\).

A railroad coal car weighs 54,600 lb empty and carries a total load of 180,000 lb of coal. The bins are equipped with bottom doors which permit discharging coal through an opening between the rails. If the car dumps coal at the rate of \(20,000 \mathrm{lb} / \mathrm{sec}\) in a downward direction relative to the car, and if frictional resistance to motion is 4 lb per ton of total remaining weight, determine the coupler force \(P\) required to give the car an acceleration of \(0.15 \mathrm{ft} / \mathrm{sec}^{2}\) in the direction of \(P\) at the instant when half the coal has been dumped.
See all solutions

Recommended explanations on Physics Textbooks

Engineering Physics

Read Explanation

Quantum Physics

Read Explanation

Electromagnetism

Read Explanation

Waves Physics

Read Explanation

Energy Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.