A `600 kg` rocket is set for a vertical firing If the exhaust speed is `100m//s` the gas ejected per sec to supply the thrust needded to overcome the weigth of rocket is .

Options

A. `117.6kg//s`
B. `58.8kg//s`
C. `6kg//s`
D. `76.4kg//s`

Explanation

To solve the problem, we need to determine the amount of gas that must be ejected per second from the rocket to generate enough thrust to overcome its weight. Here’s a step-by-step solution:

### Step 1: Identify the given data
- Mass of the rocket (m) = 600 kg
- Exhaust speed (v) = 1000 m/s
- Acceleration due to gravity (g) = 10 m/s² (assuming standard value)

### Step 2: Calculate the weight of the rocket
The weight (W) of the rocket can be calculated using the formula:
\[ W = m \cdot g \]
Substituting the known values:
\[ W = 600 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 6000 \, \text{N} \]

### Step 3: Understand the thrust required
To lift the rocket, the thrust (F) generated by the ejected gas must be equal to the weight of the rocket:
\[ F = W = 6000 \, \text{N} \]

### Step 4: Relate thrust to mass flow rate
The thrust produced by the rocket can also be expressed in terms of the mass flow rate (\( \frac{dm}{dt} \)) and the exhaust speed (v):
\[ F = v \cdot \frac{dm}{dt} \]
Where:
- \( F \) is the thrust,
- \( v \) is the exhaust speed,
- \( \frac{dm}{dt} \) is the mass flow rate of the gas.

### Step 5: Rearrange the thrust equation to find mass flow rate
We can rearrange the equation to solve for the mass flow rate:
\[ \frac{dm}{dt} = \frac{F}{v} \]

### Step 6: Substitute the known values
Now, substitute the values of thrust (F) and exhaust speed (v):
\[ \frac{dm}{dt} = \frac{6000 \, \text{N}}{1000 \, \text{m/s}} \]

### Step 7: Calculate the mass flow rate
Calculating the above expression:
\[ \frac{dm}{dt} = \frac{6000}{1000} = 6 \, \text{kg/s} \]

### Final Answer
The gas ejected per second to supply the thrust needed to overcome the weight of the rocket is **6 kg/s**.

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