A 500kg rocket is set for verticle firing. The exhaust speed is `800ms(-2)` . To give an initial upward acceleration of `20ms(-2)` , the amount of gas ejected per second to supply the needed thrust will be `(g=10ms(-2)`

Options

A. `127.5kgs^(-1)`
B. `187.5kgs^(-1)`
C. `185.5kgs^(-1)`
D. `137.5kgs^(-1)`

Explanation

To solve the problem, we need to find the amount of gas ejected per second (dm/dt) to provide the necessary thrust for the rocket to achieve an initial upward acceleration of 20 m/s².

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Mass of the rocket (m) = 500 kg
- Exhaust speed (u) = 800 m/s
- Initial upward acceleration (a) = 20 m/s²
- Acceleration due to gravity (g) = 10 m/s²

2. **Calculate the Total Thrust Required**:
The thrust (F_thrust) needed to overcome both gravity and to provide the upward acceleration can be calculated using the formula:
\[ F_{thrust} = m \cdot (a + g) \]
Substituting the values:
\[ F_{thrust} = 500 \cdot (20 + 10) = 500 \cdot 30 = 15000 \, \text{N} \]

3. **Use the Thrust Equation**:
The thrust produced by the rocket is also given by the equation:
\[ F_{thrust} = u \cdot \frac{dm}{dt} \]
Where \( \frac{dm}{dt} \) is the mass flow rate of the gas.

4. **Set the Thrust Equations Equal**:
Now we can set the two expressions for thrust equal to each other:
\[ 15000 = 800 \cdot \frac{dm}{dt} \]

5. **Solve for \( \frac{dm}{dt} \)**:
Rearranging the equation to solve for \( \frac{dm}{dt} \):
\[ \frac{dm}{dt} = \frac{15000}{800} \]
\[ \frac{dm}{dt} = 18.75 \, \text{kg/s} \]

### Final Answer:
The amount of gas ejected per second to supply the needed thrust is **18.75 kg/s**.

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