For a Rocket propulsion velocity of exhaust gases relative to rocket is 2km/s. If mass of rocket system is 1000 kg, then the rate of fuel consumption for a rockt to rise up with acceleration `4.9 m//s^(2)` will be:-

Options

A. 12.25kg/2
B. 17.5kg/s
C. 7.45 kg/s
D. 5.2kg/s

Explanation

To solve the problem, we will use the principles of Newton's laws of motion and the rocket propulsion equation. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the forces acting on the rocket
The forces acting on the rocket are:
- Thrust force (upward)
- Weight of the rocket (downward)

The thrust force can be expressed as:
\[ F_{\text{thrust}} = \text{mass flow rate} \times \text{velocity of exhaust gases} \]

### Step 2: Write down the equation of motion
According to Newton's second law, the net force acting on the rocket can be expressed as:
\[ F_{\text{net}} = F_{\text{thrust}} - mg = ma \]
Where:
- \( m \) = mass of the rocket system (1000 kg)
- \( g \) = acceleration due to gravity (approximately 10 m/s²)
- \( a \) = acceleration of the rocket (4.9 m/s²)

### Step 3: Substitute the thrust force into the equation
We can substitute the thrust force into the equation:
\[ \text{mass flow rate} \times v - mg = ma \]
Where \( v \) is the velocity of exhaust gases (2 km/s = 2000 m/s).

### Step 4: Rearrange the equation to find the mass flow rate
Rearranging the equation gives:
\[ \text{mass flow rate} \times v = mg + ma \]
\[ \text{mass flow rate} = \frac{mg + ma}{v} \]

### Step 5: Substitute the known values
Now, substituting the known values:
- \( m = 1000 \) kg
- \( g = 10 \) m/s²
- \( a = 4.9 \) m/s²
- \( v = 2000 \) m/s

We get:
\[ \text{mass flow rate} = \frac{1000 \times 10 + 1000 \times 4.9}{2000} \]

### Step 6: Calculate the mass flow rate
Calculating the right-hand side:
\[ \text{mass flow rate} = \frac{10000 + 4900}{2000} \]
\[ \text{mass flow rate} = \frac{14900}{2000} \]
\[ \text{mass flow rate} = 7.45 \, \text{kg/s} \]

### Conclusion
The rate of fuel consumption for the rocket to rise up with an acceleration of \( 4.9 \, \text{m/s}^2 \) is \( 7.45 \, \text{kg/s} \).

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