A rocket of mass 25000kg is cruising through space with a constant speed of 1000m/s^-1 when exhaust gases are expelled for 10 seconds at a rate of 500kg/s^-1 with a speed of 50000m/s^-1. Find the new speed of the rocket

To find the new speed of the rocket after the exhaust gases are expelled, we can use the principle of conservation of momentum. Here's how to approach the problem:

### Given:
- Mass of the rocket before expelling gases, \( m_0 = 25000 \, \text{kg} \)
- Initial speed of the rocket, \( v_0 = 1000 \, \text{m/s} \)
- Rate of mass ejection, \( \dot{m} = 500 \, \text{kg/s} \)
- Velocity of exhaust gases relative to the rocket, \( v_e = 50000 \, \text{m/s} \)
- Time for which gases are expelled, \( t = 10 \, \text{seconds} \)

### Step 1: Calculate the total mass ejected
The total mass ejected during the 10 seconds is:
\[
\Delta m = \dot{m} \times t = 500 \, \text{kg/s} \times 10 \, \text{seconds} = 5000 \, \text{kg}
\]

### Step 2: Calculate the final mass of the rocket
After the gases are expelled, the final mass of the rocket is:
\[
m_f = m_0 - \Delta m = 25000 \, \text{kg} - 5000 \, \text{kg} = 20000 \, \text{kg}
\]

### Step 3: Apply conservation of momentum
Since momentum is conserved, the initial momentum of the system (rocket + exhaust gases) should be equal to the final momentum of the system.

Initial momentum:
\[
p_{\text{initial}} = m_0 \times v_0 = 25000 \, \text{kg} \times 1000 \, \text{m/s} = 25000000 \, \text{kg} \cdot \text{m/s}
\]

Final momentum:
\[
p_{\text{final}} = m_f \times v_f + \Delta m \times v_e
\]
where \( v_f \) is the final velocity of the rocket.

Substituting the values:
\[
25000000 \, \text{kg} \cdot \text{m/s} = 20000 \, \text{kg} \times v_f + 5000 \, \text{kg} \times 50000 \, \text{m/s}
\]
\[
25000000 = 20000v_f + 250000000
\]

### Step 4: Solve for the final velocity \( v_f \)
Rearranging the equation to solve for \( v_f \):
\[
20000v_f = 25000000 - 250000000
\]
\[
20000v_f = -225000000
\]
\[
v_f = \frac{-225000000}{20000} = -11250 \, \text{m/s}
\]

### Final Answer:
The negative sign indicates that the rocket's direction has changed, and the new speed of the rocket is \( 11250 \, \text{m/s} \) in the opposite direction to its original motion.

Would you like a more detailed explanation of any part? Here are some additional questions related to this topic:

1. What is the principle of conservation of momentum, and how does it apply to rocket propulsion?
2. How would the final speed change if the exhaust velocity were different?
3. What would happen if the rocket expelled gases for a longer time?
4. Can you explain how the direction of the rocket's velocity changes after expelling gases?
5. What would be the rocket's final velocity if no gases were expelled?

**Tip:** In rocket propulsion, understanding the relationship between mass, velocity, and momentum is key to predicting changes in speed and direction.

Calculate the new speed of a rocket after expelling gases at a specific rate and velocity using the principle of conservation of momentum. Understand how mass ejection affects rocket velocity. Suitable for advanced high school physics students.

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