A 1000kg rocket is set for vertical firing. The exhaust speed is `500m//s`.To give the rocket an initial upward acceleration of `30 m//s^(2)`, the amount of gas ejected per second to supply the needed thrust is

Options

A. 80 kg `//` sec
B. 40 kg `//` sec
C. 30 kg `//` sec
D. 10 kg `//` sec

Answer

To solve the problem, we need to determine the amount of gas (mass) ejected per second (dm/dt) that is required to provide the necessary thrust for the rocket to achieve the desired upward acceleration.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the rocket (m) = 1000 kg
- Exhaust speed (V) = 500 m/s
- Upward acceleration (a) = 30 m/s²
- Gravitational acceleration (g) = 9.81 m/s² (approximately)

2. **Calculate the Weight of the Rocket:**
The weight (W) of the rocket can be calculated using the formula:
\[ W = m \cdot g = 1000 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 9810 \, \text{N} \]

3. **Determine the Required Thrust Force:**
According to Newton's second law, the net force acting on the rocket is given by:
\[ F_{\text{net}} = F_{\text{thrust}} - W \]
Rearranging this gives:
\[ F_{\text{thrust}} = F_{\text{net}} + W \]
The net force (F_net) required to achieve the upward acceleration can be calculated as:
\[ F_{\text{net}} = m \cdot a = 1000 \, \text{kg} \cdot 30 \, \text{m/s}^2 = 30000 \, \text{N} \]
Therefore, the thrust force required is:
\[ F_{\text{thrust}} = 30000 \, \text{N} + 9810 \, \text{N} = 39810 \, \text{N} \]

4. **Relate Thrust Force to Mass Ejection Rate:**
The thrust force can also be expressed in terms of the exhaust speed and the rate of mass ejection:
\[ F_{\text{thrust}} = V \cdot \frac{dm}{dt} \]
Substituting the values we have:
\[ 39810 \, \text{N} = 500 \, \text{m/s} \cdot \frac{dm}{dt} \]

5. **Solve for the Mass Ejection Rate (dm/dt):**
Rearranging the equation to find dm/dt gives:
\[ \frac{dm}{dt} = \frac{39810 \, \text{N}}{500 \, \text{m/s}} = 79.62 \, \text{kg/s} \]
Rounding this to two decimal places, we get:
\[ \frac{dm}{dt} \approx 80 \, \text{kg/s} \]

### Final Answer:
The amount of gas ejected per second to supply the needed thrust is approximately **80 kg/s**.