A rocket of mass 6000kg is set for vertical firing. If the exhaust speed is 1000m/sec. The rate of mass of gas ejected to supply a thrust needed to give the rocket an initial upward acceleration of 20.2 m/ `s^2` is

Options

A. 150 kg/sec
B. 160 kg/sec
C. 28 kg/sec
D. 180 kg/sec

Answer

To solve the problem, we need to determine the rate of mass of gas ejected (dm/dt) required to provide the necessary thrust to give the rocket an initial upward acceleration of 20.2 m/s².

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the rocket (m) = 6000 kg
- Exhaust speed (v) = 1000 m/s
- Acceleration (a) = 20.2 m/s²
- Gravitational acceleration (g) = 9.81 m/s² (approximately)

2. **Calculate the Total Force Required (Thrust):**
The thrust (F) needed to accelerate the rocket can be calculated using Newton's second law:
\[ F = m(g + a) \]
Substituting the values:
\[ F = 6000 \, \text{kg} \times (9.81 \, \text{m/s}^2 + 20.2 \, \text{m/s}^2) \]
\[ F = 6000 \, \text{kg} \times 30.01 \, \text{m/s}^2 \]
\[ F = 180060 \, \text{N} \]

3. **Relate Thrust to Mass Ejection Rate:**
The thrust can also be expressed in terms of the rate of mass ejection (dm/dt) and the exhaust speed (v):
\[ F = \frac{dm}{dt} \times v \]
Rearranging this gives:
\[ \frac{dm}{dt} = \frac{F}{v} \]

4. **Substituting the Values:**
Now we can substitute the thrust and exhaust speed into the equation:
\[ \frac{dm}{dt} = \frac{180060 \, \text{N}}{1000 \, \text{m/s}} \]
\[ \frac{dm}{dt} = 180.06 \, \text{kg/s} \]

5. **Final Result:**
The rate of mass of gas ejected to supply the thrust needed is approximately:
\[ \frac{dm}{dt} \approx 180.06 \, \text{kg/s} \]

### Summary:
The rate of mass of gas ejected to supply the thrust needed for the rocket's initial upward acceleration of 20.2 m/s² is approximately **180.06 kg/s**.