A 600 kg rocket is set for a vertical firing. If the exhaust speed is 1000 `ms^(-1)`, the mass of the gas ejected per second to supply the thrust needed to overcome the weight of rocket is

Options

A. 117.6 kg `s^(-1)`
B. 58.6 kg `s^(-1)`
C. 6 kg `s^(-1)`
D. 76.4 kg `s^(-1)`

Answer

To solve the problem, we need to find the mass of the gas ejected per second (dm/dt) required to provide the thrust needed to overcome the weight of the rocket. Here’s how we can approach this step by step:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the rocket (m) = 600 kg
- Exhaust speed (u) = 1000 m/s
- Acceleration due to gravity (g) = 10 m/s² (standard value)

2. **Calculate the Weight of the Rocket:**
- The weight (W) of the rocket can be calculated using the formula:
\[ W = m \cdot g \]
- Substituting the values:
\[ W = 600 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 6000 \, \text{N} \]

3. **Understand the Thrust Requirement:**
- To lift the rocket, the thrust (F) produced by the exhaust gases must be equal to the weight of the rocket. Therefore:
\[ F = W = 6000 \, \text{N} \]

4. **Use the Thrust Equation:**
- The thrust produced by the rocket can be expressed as:
\[ F = u \cdot \frac{dm}{dt} \]
- Where \( \frac{dm}{dt} \) is the mass flow rate of the gas.

5. **Rearranging the Equation:**
- We can rearrange the equation to find \( \frac{dm}{dt} \):
\[ \frac{dm}{dt} = \frac{F}{u} \]

6. **Substituting the Values:**
- Now, substituting the values of F and u:
\[ \frac{dm}{dt} = \frac{6000 \, \text{N}}{1000 \, \text{m/s}} = 6 \, \text{kg/s} \]

7. **Final Result:**
- The mass of the gas ejected per second is:
\[ \frac{dm}{dt} = 6 \, \text{kg/s} \]

### Conclusion:
The mass of the gas ejected per second to supply the thrust needed to overcome the weight of the rocket is **6 kg/s**.