To determine the fraction of the initial mass of the spacecraft that must be discarded as exhaust during a midcourse correction, we can use the Tsiolkovsky rocket equation, which relates the change in velocity of a rocket to its mass and the exhaust velocity of its engines. Let's break this down step by step.
Understanding the Rocket Equation
The Tsiolkovsky rocket equation is given by:
Δv = v_e * ln(m_0 / m_f)
Where:
- Δv = change in velocity (in this case, 22.6 m/s)
- v_e = exhaust velocity (1230 m/s)
- m_0 = initial mass of the spacecraft
- m_f = final mass of the spacecraft after burning fuel
Setting Up the Equation
We need to rearrange the equation to find the mass ratio:
m_0 / m_f = e^(Δv / v_e)
First, we can calculate the mass ratio:
Substituting the values:
m_0 / m_f = e^(22.6 / 1230)
Calculating the Exponential
Now, let's compute the exponent:
22.6 / 1230 ≈ 0.01835
Now, we can find the exponential:
e^(0.01835) ≈ 1.01854
This means:
m_0 / m_f ≈ 1.01854
Finding the Mass Discarded
Next, we can express the final mass in terms of the initial mass:
m_f = m_0 / 1.01854
The mass discarded (the mass of the fuel used) can be calculated as:
mass discarded = m_0 - m_f = m_0 - (m_0 / 1.01854)
Factoring this out gives:
mass discarded = m_0 * (1 - 1 / 1.01854)
Calculating the Fraction of Mass Discarded
Now, let's find the fraction of the initial mass that is discarded:
fraction discarded = 1 - 1 / 1.01854 ≈ 0.0177
This means that approximately 1.77% of the initial mass of the spacecraft must be discarded as exhaust to achieve the required midcourse correction of 22.6 m/s.
Summary of Results
In summary, to achieve a midcourse correction of 22.6 m/s with an exhaust speed of 1230 m/s, the spacecraft needs to discard about 1.77% of its initial mass as exhaust. This calculation illustrates the efficiency of rocket propulsion and the importance of mass management in space missions.
