A spaceship's orbital maneuver requires a speed increase of 1.45 ✕ 103 m/s. If its engine has an exhaust speed of 2.45 ✕ 103 m/s,determine the required ratio

*Mi*

*Mf*

of its initial mass to its final mass. (The difference

*Mi* − *Mf*

equals the mass of the ejected fuel.)

The required ratio of the initial mass \( M_i \) to the final mass \( M_f \) can be found using the **Tsiolkovsky rocket equation**:

\[
\Delta v = v_e \ln \left( \frac{M_i}{M_f} \right)
\]

where:
- \( \Delta v \) is the change in velocity (1.45 ✕ 10³ m/s),
- \( v_e \) is the exhaust velocity (2.45 ✕ 10³ m/s),
- \( \ln \) is the natural logarithm.

### Step 1: Substitute the given values
Plugging in the values, we get:

\[
1.45 \times 10^3 = 2.45 \times 10^3 \ln \left( \frac{M_i}{M_f} \right)
\]

### Step 2: Isolate \( \ln \left( \frac{M_i}{M_f} \right) \)
Divide both sides by \( v_e = 2.45 \times 10^3 \):

\[
\ln \left( \frac{M_i}{M_f} \right) = \frac{1.45 \times 10^3}{2.45 \times 10^3}
\]

Calculating the right side:

\[
\ln \left( \frac{M_i}{M_f} \right) \approx 0.5918
\]

### Step 3: Exponentiate to solve for \( \frac{M_i}{M_f} \)
To find \( \frac{M_i}{M_f} \), take the exponential of both sides:

\[
\frac{M_i}{M_f} = e^{0.5918}
\]

Calculating this:

\[
\frac{M_i}{M_f} \approx 1.807
\]

### Conclusion
The required mass ratio \( \frac{M_i}{M_f} \) is approximately **1.807**.

---

Would you like further details on any steps?

1. What is the Tsiolkovsky rocket equation, and how is it derived?
2. How does exhaust velocity affect the fuel requirements for maneuvers?
3. What are real-world applications of the Tsiolkovsky rocket equation?
4. How is natural logarithm used in physics, especially in exponential growth and decay?
5. Why is the mass ratio significant in determining the efficiency of rocket propulsion?

**Tip:** The higher the exhaust velocity, the less fuel mass is needed for the same velocity change, making propulsion more efficient.

A spaceship's orbital maneuver requires a speed increase of 1.45 ✕ 10³ m/s. If its engine has an exhaust speed of 2.45 ✕ 10³ m/s, determine the required ratio Mi/Mf of its initial mass to its final mass. (The difference Mi − Mf equals the mass of the ejected fuel.)

Learn to calculate the required mass ratio Mi/Mf for a spaceship's orbital maneuver requiring a speed increase of 1.45 ✕ 10³ m/s, with an exhaust speed of 2.45 ✕ 10³ m/s. This problem applies the Tsiolkovsky rocket equation, involving natural logarithms and rocket propulsion principles. Suitable for college-level physics students.

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