/*! 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 169 A large two-stage liquid rocket ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 169

A large two-stage liquid rocket with mass of \(30,000 \mathrm{kg}\) is to be launched from a sea-level launch pad. The main engine burns liquid hydrogen and liquid oxygen in a stoichiometric mixture at \(2450 \mathrm{kg} / \mathrm{s}\). The thrust nozzle has an exit diameter of \(2.6 \mathrm{m}\). The exhaust gases exit the nozale at \(2270 \mathrm{m} / \mathrm{s}\) and an exit plane pressure of \(66 \mathrm{kPa}\) absolute. Calculate the acceleration of the rocket at liftoff. Obtain an expression for speed as a function of time, neglecting air resistance.

Short Answer

Expert verified
The acceleration of the rocket at liftoff is \(175.7 \, m/s^2\) and the speed as a function of time is \( v(t) = 175.7 \cdot t \)

Step by step solution

01

Determine the thrust force

The rocket's main engine burns fuel at a rate of 2450 kg/s and the exhaust gases exit the nozzle at a velocity of 2270 m/s. This creates a thrust force which can be calculated using the formula \( F_t=m \cdot v \), where \(m\) is the mass flow rate of fuel and \(v\) is the exhaust velocity. So, \( F_t=2450 \cdot 2270=5565000 \, N \)
02

Calculate net force

The net force acting on the rocket at liftoff includes the thrust from the engine as well as the force of gravity. Gravity exerts a downward force equal to the mass of the rocket multiplied by the acceleration due to gravity, which is approximately 9.8 m/s\(^2\). Hence, the gravitational force is \(F_g=30000 \cdot 9.8=294000 \, N\). The net force is then the difference between the thrust and the gravitational force, i.e. \( F_{net}=F_t - F_g=5565000 - 294000=5271000 \, N \)
03

Calculate acceleration

The acceleration of the rocket can be calculated using Newton's Second Law, which states that the acceleration is equal to the net force divided by the mass of the object. Hence, the acceleration is \( a=F_{net}/m=5271000/30000=175.7 \, m/s^2 \)
04

Obtain speed as a function of time

Finally, recall that under constant acceleration, the speed of an object as a function of time is given by \( v=a \cdot t \), where \( v \) is the speed, \( a \) is the acceleration and \( t \) is the time. Hence, the speed of the rocket as a function of time is \( v(t) = 175.7 \cdot t \)

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