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A spacecraft has a dry mass of 197 kg, and carries 35 kg of hydrazine propellant. It is equipped with a set of six thrusters, each with an Isp of 300 s. How much propellant must it expel to achieve a Δv of 212 m/s?

A) 11.55 kg
B) 16.10 kg
C) 16.80 kg
D) 3.85 kg

Answer :

To solve this question, we need to use the Tsiolkovsky rocket equation, which relates the change in velocity (Δv) of a spacecraft to the mass of propellant expelled.

The rocket equation is given by:

[tex]\Delta v = I_{sp} \cdot g \cdot \ln\left(\frac{m_i}{m_f}\right)[/tex]

Where:

  • [tex]\Delta v[/tex] is the change in velocity.
  • [tex]I_{sp}[/tex] is the specific impulse of the thrusters, given as 300 s.
  • [tex]g[/tex] is the acceleration due to gravity, approximately 9.81 m/s² on Earth's surface.
  • [tex]m_i[/tex] is the initial total mass of the spacecraft.
  • [tex]m_f[/tex] is the final total mass of the spacecraft after the propellant is burnt.

Given:

  • Dry mass of the spacecraft = 197 kg.
  • Mass of hydrazine propellant = 35 kg.
  • [tex]\Delta v = 212[/tex] m/s.

We need to find how much propellant must be expelled, which involves finding [tex]m_f[/tex].

Initial Setup

The initial mass [tex]m_i[/tex] is the total mass when the spacecraft is fully loaded with propellant:

[tex]m_i = 197 + 35 = 232 \text{ kg}[/tex]

Rearranging the Equation

Rearrange the Tsiolkovsky rocket equation to solve for [tex]m_f[/tex]:

[tex]\Delta v = I_{sp} \cdot g \cdot \ln\left(\frac{m_i}{m_f}\right)[/tex]

[tex]212 = 300 \times 9.81 \times \ln\left(\frac{232}{m_f}\right)[/tex]

Divide both sides by [tex]300 \times 9.81[/tex]:

[tex]\ln\left(\frac{232}{m_f}\right) = \frac{212}{2943} \approx 0.072[/tex]

Solving for [tex]\frac{232}{m_f}[/tex]:

[tex]\frac{232}{m_f} = e^{0.072} \approx 1.07481[/tex]

Thus, solve for [tex]m_f[/tex]:

[tex]m_f = \frac{232}{1.07481} \approx 215.949 \text{ kg}[/tex]

Finding the Propellant Expelled

The propellant expelled is the difference between the initial and final mass:

[tex]\text{Propellant expelled} = m_i - m_f = 232 - 215.949 \approx 16.051 \text{ kg}[/tex]

Closest option from the choices given is:

B) 16.10 kg

Therefore, the selected answer is Option B: 16.10 kg.

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