/*! 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 162 A rocket sled accelerates from r... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 162

A rocket sled accelerates from rest on a level track with negligible air and rolling resistances. The initial mass of the sled is \(M_{0}=600 \mathrm{kg}\). The rocket initially contains \(150 \mathrm{kg}\) of fuel. The rocket motor burns fuel at constant rate \(m=15 \mathrm{kg} / \mathrm{s}\). Exhaust gases leave the rocket nozzle uniformly and axially at \(V_{c}=2900 \mathrm{m} / \mathrm{s}\) relative to the nozzle, and the pressure is atmospheric. Find the maximum speed reached by the rocket sled. Calculate the maximum acceleration of the sled during the run.

Short Answer

Expert verified
The maximum speed reached by the rocket sled is 725 m/s and the maximum acceleration is 72.5 m/s虏.

Step by step solution

01

Establish the Thrust of the Rocket

The thrust (\( F \)) of the rocket is calculated using the formula \( F = m \cdot Vc \), where \( m \) is the rate of fuel burn and \( Vc \) is the exhaust gas velocity. Substituting the given values, we find \( F = 15 \, kg/s \cdot 2900 \, m/s = 43500 \, N \).
02

Determine the Burn Time

The burn time (\( t \)) can be calculated by dividing the total amount of fuel by the rate of fuel burn: \( t = 150 \, kg / 15 \, kg/s = 10 \, s \).
03

Calculate the Maximum Speed

Using the equation of motion \( v = u + at \), where \( u \) is the initial velocity (0 in this case), \( a \) is acceleration and \( t \) is time, we first need to establish the acceleration. Acceleration (\( a \)) is given by the formula \( a = F/M_0 \), substituting the values, we find \( a = 43500 \, N / 600 \, kg = 72.5 \, m/s^2 \). Now the maximum speed (\( v_{max} \)), is given by: \( v_{max} = 0 + 72.5 \, m/s^2 \cdot 10 \, s = 725 \, m/s \). The maximum speed of the rocket sled is 725 m/s.
04

Calculate the Maximum Acceleration

Acceleration decreases as the rocket burns fuel and its mass decreases. However, the maximum acceleration occurs when the mass of the rocket is greatest, which is at the start of the burn (since it's carrying all the fuel). Thus, the maximum acceleration is the value we calculated in Step 3, \( 72.5 \, m/s^2 \).

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Most popular questions from this chapter

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