A rocket of mass 100 kg burns 0.1 kg of fuel per sec. If velocity of exhaust gas is 1 km/sec, then it lifts with an acceleration of

Options

A. `1000 ms^(-2)`
B. `100 ms^(-2)`
C. `10 ms^(-2)`
D. `1 ms^(-2)`

Explanation

To solve the problem of the rocket's acceleration, we can use the principles of momentum and thrust. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Given Data
- Mass of the rocket (m) = 100 kg
- Rate of fuel consumption (dm/dt) = 0.1 kg/s
- Velocity of exhaust gas (v) = 1 km/s = 1000 m/s (conversion from km/s to m/s)

### Step 2: Understand the Thrust Force
The thrust force (Ft) generated by the rocket can be calculated using the formula:
\[ F_t = v \cdot \frac{dm}{dt} \]
Where:
- \( v \) is the velocity of the exhaust gas.
- \( \frac{dm}{dt} \) is the rate of change of mass (mass flow rate of the exhaust).

### Step 3: Calculate the Thrust Force
Substituting the known values into the thrust formula:
\[ F_t = 1000 \, \text{m/s} \cdot 0.1 \, \text{kg/s} \]
\[ F_t = 100 \, \text{N} \]

### Step 4: Apply Newton's Second Law
According to Newton's second law, the net force acting on the rocket is equal to the mass of the rocket multiplied by its acceleration (a):
\[ F_{\text{net}} = m \cdot a \]
The net force acting on the rocket is the thrust force minus the weight of the rocket (W):
\[ F_{\text{net}} = F_t - W \]
Where weight \( W = m \cdot g \) (g = acceleration due to gravity, approximately \( 9.81 \, \text{m/s}^2 \)):
\[ W = 100 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 981 \, \text{N} \]

### Step 5: Set Up the Equation
Now, we can set up the equation:
\[ F_t - W = m \cdot a \]
Substituting the values:
\[ 100 \, \text{N} - 981 \, \text{N} = 100 \, \text{kg} \cdot a \]
\[ -881 \, \text{N} = 100 \, \text{kg} \cdot a \]

### Step 6: Solve for Acceleration (a)
To find the acceleration, rearrange the equation:
\[ a = \frac{-881 \, \text{N}}{100 \, \text{kg}} \]
\[ a = -8.81 \, \text{m/s}^2 \]

### Conclusion
The negative sign indicates that the thrust is not sufficient to lift the rocket against gravity; thus, the rocket does not accelerate upwards. Instead, it experiences a downward acceleration of \( 8.81 \, \text{m/s}^2 \).