A 5000 kg rocket is set for vertical firing. The exhaust speed is `800 ms^(-1)`.To give an intial upward acceleration of 20 `m//s^(2)` , the amount of gas ejected per second to supply the needed thrust will be (`g=10ms^(-2)`)

Options

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

Answer

To solve the problem, we need to determine the amount of gas ejected per second (dm/dt) to provide the necessary thrust for the rocket.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the rocket (m) = 5000 kg
- Exhaust speed (V) = 800 m/s
- Initial upward acceleration (a) = 20 m/s²
- Acceleration due to gravity (g) = 10 m/s²

2. **Calculate the Weight of the Rocket:**
The weight (W) of the rocket can be calculated using the formula:
\[ W = m \cdot g = 5000 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 50000 \, \text{N} \]

3. **Calculate the Required Thrust Force:**
The thrust force (F_thrust) needed to achieve the upward acceleration can be found using Newton's second law:
\[ F_{\text{thrust}} = W + m \cdot a \]
Substituting the values:
\[ F_{\text{thrust}} = 50000 \, \text{N} + (5000 \, \text{kg} \cdot 20 \, \text{m/s}^2) = 50000 \, \text{N} + 100000 \, \text{N} = 150000 \, \text{N} \]

4. **Use the Thrust Equation:**
The thrust force can also be expressed in terms of the mass flow rate (dm/dt) and exhaust speed (V):
\[ F_{\text{thrust}} = V \cdot \frac{dm}{dt} \]
Rearranging this gives us:
\[ \frac{dm}{dt} = \frac{F_{\text{thrust}}}{V} \]

5. **Substituting the Values:**
Now, substituting the thrust force and exhaust speed into the equation:
\[ \frac{dm}{dt} = \frac{150000 \, \text{N}}{800 \, \text{m/s}} = 187.5 \, \text{kg/s} \]

6. **Final Answer:**
The amount of gas ejected per second to supply the needed thrust is:
\[ \frac{dm}{dt} = 187.5 \, \text{kg/s} \]