A rocket of mass 1000 kg is to be projected vertically upwards. The gases are exhausted vertically downwards with velocity `100m//s` with respect to the rocket. (What is the minimum rate of burning fuel, so as to just lift the rocket upwards against the gravitational attraction?) (Take `g=10m//s^(2)`)

Options

A. 50 kg/s
B. 100kg/s
C. 200kg/s
D. 400kg/s

Explanation

To solve the problem of determining the minimum rate of burning fuel for the rocket to just lift off against gravitational attraction, we can follow these steps:

### Step 1: Understand the Forces Acting on the Rocket
The rocket must generate enough thrust to overcome the gravitational force acting on it. The gravitational force \( F_g \) can be calculated using the formula:
\[ F_g = m \cdot g \]
where:
- \( m \) is the mass of the rocket (1000 kg)
- \( g \) is the acceleration due to gravity (10 m/s²)

### Step 2: Calculate the Gravitational Force
Substituting the values into the formula:
\[ F_g = 1000 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 10000 \, \text{N} \]
This means the rocket needs to exert a thrust of at least 10000 N to just lift off.

### Step 3: Relate Thrust to Mass Flow Rate
The thrust \( F \) produced by the rocket can also be expressed in terms of the mass flow rate \( \frac{dm}{dt} \) and the exhaust velocity \( v_e \):
\[ F = \frac{dm}{dt} \cdot v_e \]
where:
- \( v_e \) is the velocity of the exhaust gases (100 m/s)

### Step 4: Set Up the Equation for Thrust
To find the minimum rate of burning fuel, we set the thrust equal to the gravitational force:
\[ \frac{dm}{dt} \cdot v_e = F_g \]
Substituting the known values:
\[ \frac{dm}{dt} \cdot 100 \, \text{m/s} = 10000 \, \text{N} \]

### Step 5: Solve for the Mass Flow Rate
Now, we can solve for \( \frac{dm}{dt} \):
\[ \frac{dm}{dt} = \frac{10000 \, \text{N}}{100 \, \text{m/s}} = 100 \, \text{kg/s} \]

### Conclusion
The minimum rate of burning fuel required for the rocket to just lift off is:
\[ \frac{dm}{dt} = 100 \, \text{kg/s} \]

### Final Answer
Thus, the answer is **100 kg/s**.

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