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Chapter 4: Problem 164

A rocket sled with initial mass of \(900 \mathrm{kg}\) is to be accelerated on a level track. The rocket motor burns fuel at constant rate \(m=13.5 \mathrm{kg} / \mathrm{s}\). The rocket exhaust flow is uniform and axial. Gases leave the nozzle at \(2750 \mathrm{m} / \mathrm{s}\) relative to the nozzle, and the pressure is atmospheric. Determine the minimum mass of rocket fuel needed to propel the sled to a speed of \(265 \mathrm{m} / \mathrm{s}\) before burnout occurs. As a first approximation, neglect resistance forces.

Short Answer

Expert verified
The minimum mass of rocket fuel needed is obtained by calculating the difference between the given initial mass of the rocket sled and the final mass obtained using the rocket equation and the given parameters. Plug in the given values and solve to obtain the final answer.

Step by step solution

01

Identify the Given Values

First, identify and write down the given values in the problem: the initial mass of the rocket sled \(m_{0} = 900 \mathrm{kg}\), the fuel burn rate \(m = 13.5 \mathrm{kg/s}\), the exhaust speed \(v_{e} = 2750 \mathrm{m/s}\), and the final speed required \(v = 265 \mathrm{m/s}\).
02

Apply the Rocket Equation

Next, apply the rocket equation to find the change in mass required to achieve the desired final speed. The rocket equation is \(v = v_e * ln(m_0/m_f)\), where \(v\) is the final velocity, \(v_e\) is exhaust velocity, \(\ln\) is the natural logarithm, \(m_0\) is the total initial mass and \(m_f\) is the final mass.
03

Solve for the Final Mass

Rearrange the equation from step 2 in order to solve for \(m_f\). Doing this gives us the equation \(m_f = m_0 / e^{(v/v_{e})}\).
04

Determine the Amount of Fuel Consumed

The amount of fuel consumed is given by the difference between the initial mass and the final mass, i.e., \(\Delta m = m_{0} - m_f\). Plug in the calculated value of \(m_f\) from step 3 and the given value of \(m_0\) here to get the required amount of fuel.

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