A 6090 kg space probe moving nose-first toward Jupiter at 120 m/s relative to the Sun fires its rocket engine, ejecting 70.0 kg of exhaust at a speed of 253 m/s relative to the space probe. What is the final velocity of the probe?

Answer

To solve the problem of finding the final velocity of the space probe after it ejects exhaust, we will use the principle of conservation of momentum. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Given Information
- Mass of the space probe (m₁) = 6090 kg
- Initial velocity of the space probe (u₁) = 120 m/s (toward Jupiter)
- Mass of the exhaust (m₂) = 70 kg
- Speed of the exhaust relative to the space probe (v₂') = 253 m/s (in the opposite direction)

### Step 2: Calculate the Speed of the Exhaust Relative to the Sun
The speed of the exhaust relative to the Sun (v₂) can be calculated by adding the speed of the exhaust relative to the probe to the speed of the probe itself. Since the exhaust is ejected in the opposite direction to the probe's motion, we will subtract the speed of the probe from the speed of the exhaust.

\[ v₂ = u₁ - v₂' = 120 \, \text{m/s} - 253 \, \text{m/s} = -133 \, \text{m/s} \]

### Step 3: Apply Conservation of Momentum
The total momentum before the exhaust is ejected must equal the total momentum after the exhaust is ejected.

**Initial Momentum (P_initial):**
\[ P_{\text{initial}} = m₁ \cdot u₁ = 6090 \, \text{kg} \cdot 120 \, \text{m/s} = 730800 \, \text{kg m/s} \]

**Final Momentum (P_final):**
After the exhaust is ejected, the mass of the probe becomes (m₁ - m₂) and its final velocity is v. The momentum of the exhaust can be calculated as m₂ times its velocity v₂.

\[ P_{\text{final}} = (m₁ - m₂) \cdot v + m₂ \cdot v₂ \]
\[ P_{\text{final}} = (6090 \, \text{kg} - 70 \, \text{kg}) \cdot v + 70 \, \text{kg} \cdot (-133 \, \text{m/s}) \]

### Step 4: Set Initial Momentum Equal to Final Momentum
Setting the initial momentum equal to the final momentum gives us:

\[ 730800 \, \text{kg m/s} = (6020 \, \text{kg}) \cdot v + 70 \, \text{kg} \cdot (-133 \, \text{m/s}) \]

Calculating the momentum of the exhaust:

\[ 70 \cdot (-133) = -9310 \, \text{kg m/s} \]

Now substituting back into the equation:

\[ 730800 = 6020 \cdot v - 9310 \]

### Step 5: Solve for Final Velocity (v)
Rearranging the equation to solve for v:

\[ 6020 \cdot v = 730800 + 9310 \]
\[ 6020 \cdot v = 739110 \]
\[ v = \frac{739110}{6020} \approx 122.8 \, \text{m/s} \]

### Step 6: Conclusion
The final velocity of the space probe after ejecting the exhaust is approximately **122.8 m/s** toward Jupiter.

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