A 5000 kg rocket is set for verticle firing the relative speed of burnt gas is `800ms^(-1)` To give an initial upwards acceleration of `20ms^(-2)` the amount of gas ejected per second to supply the needed thrust will be

Options

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

Explanation

To solve the problem, we need to determine the amount of gas that must be ejected per second to provide the necessary thrust for the rocket to achieve the desired acceleration. We will use Newton's second law and the concept of thrust in rocket propulsion.

### Step-by-Step Solution:

1. **Identify the forces acting on the rocket:**
The forces acting on the rocket include the thrust force (upward) and the weight of the rocket (downward). The weight can be calculated using the formula:
\[ W = m \cdot g \]
where \( m = 5000 \, \text{kg} \) (mass of the rocket) and \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity).

\[ W = 5000 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 49050 \, \text{N} \]

2. **Calculate the total force required for the upward acceleration:**
According to Newton's second law, the net force \( F_{\text{net}} \) required to achieve an upward acceleration \( a \) can be expressed as:
\[ F_{\text{net}} = m \cdot a \]
where \( a = 20 \, \text{m/s}^2 \).

\[ F_{\text{net}} = 5000 \, \text{kg} \cdot 20 \, \text{m/s}^2 = 100000 \, \text{N} \]

3. **Determine the thrust force required:**
The thrust force \( F_{\text{thrust}} \) must overcome both the weight of the rocket and provide the necessary net force for acceleration:
\[ F_{\text{thrust}} = F_{\text{net}} + W \]
\[ F_{\text{thrust}} = 100000 \, \text{N} + 49050 \, \text{N} = 149050 \, \text{N} \]

4. **Use the thrust equation to find the mass flow rate of the gas:**
The thrust produced by the rocket can also be expressed in terms of the mass flow rate \( \dot{m} \) of the gas and the relative speed \( v_e \) of the burnt gases:
\[ F_{\text{thrust}} = \dot{m} \cdot v_e \]
Rearranging this gives us the mass flow rate:
\[ \dot{m} = \frac{F_{\text{thrust}}}{v_e} \]
Given \( v_e = 800 \, \text{m/s} \):
\[ \dot{m} = \frac{149050 \, \text{N}}{800 \, \text{m/s}} = 186.3125 \, \text{kg/s} \]

5. **Final Result:**
The amount of gas ejected per second to supply the needed thrust is approximately:
\[ \dot{m} \approx 186.31 \, \text{kg/s} \]