A 7500 kg rocket is set for vertical firing. The exhaust speed is 800 m/s w.r.to rocket. To give an vertical upward acceleration of `10m//s^(2)`, the amount of gas ejected per second for the needed thrust will be

Options

A. 127.5 kg/s
B. 137.5 kg/s
C. 185.5 kg/s
D. 187.5 kg/s

Answer

To solve the problem, we need to determine the amount of gas ejected per second (dm/dt) required for the rocket to achieve a vertical upward acceleration of 10 m/s². We will use the principles of thrust and Newton's second law of motion.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the rocket (m) = 7500 kg
- Exhaust speed (v) = 800 m/s
- Acceleration (a) = 10 m/s²
- Gravitational acceleration (g) = 9.81 m/s² (approximately)

2. **Calculate the Total Force Required (Thrust):**
The thrust force (F_thrust) required to lift the rocket must overcome both the gravitational force and provide the necessary upward acceleration. This can be expressed as:
\[ F_{thrust} = mg + ma \]
Where:
- \( mg \) is the weight of the rocket.
- \( ma \) is the force needed for the upward acceleration.

Plugging in the values:
\[ F_{thrust} = (7500 \, \text{kg} \times 9.81 \, \text{m/s}^2) + (7500 \, \text{kg} \times 10 \, \text{m/s}^2) \]
\[ F_{thrust} = 73575 \, \text{N} + 75000 \, \text{N} = 148575 \, \text{N} \]

3. **Relate Thrust to Mass Flow Rate:**
The thrust can also be expressed in terms of the mass flow rate of the exhaust gases (dm/dt) and the exhaust speed (v):
\[ F_{thrust} = v \cdot \frac{dm}{dt} \]
Rearranging this equation gives us:
\[ \frac{dm}{dt} = \frac{F_{thrust}}{v} \]

4. **Substitute the Values:**
Now, substitute the thrust force and exhaust speed into the equation:
\[ \frac{dm}{dt} = \frac{148575 \, \text{N}}{800 \, \text{m/s}} \]
\[ \frac{dm}{dt} = 185.71875 \, \text{kg/s} \]

5. **Final Answer:**
Rounding to three significant figures, the amount of gas ejected per second is approximately:
\[ \frac{dm}{dt} \approx 185.72 \, \text{kg/s} \]

### Summary:
The amount of gas ejected per second for the needed thrust is approximately **185.72 kg/s**.