A 1000kg rocket is set for vertical firing. The exhaust speed is `500m//s`.To give the rocket an initial upward acceleration of `30 m//s^(2)`, the amount of gas ejected per second to supply the needed thrust is

Options

A. 80 kg `//` sec
B. 40 kg `//` sec
C. 30 kg `//` sec
D. 10 kg `//` sec

Answer

To solve the problem, we need to determine the amount of gas ejected per second (dm/dt) required to provide the necessary thrust for the rocket. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the forces acting on the rocket
The forces acting on the rocket are:
1. The weight of the rocket (W = mg)
2. The thrust force (T) produced by the ejection of gas.

### Step 2: Write the thrust force equation
The thrust force can be expressed in terms of the exhaust speed (v) and the mass flow rate (dm/dt):
\[ T = v \cdot \frac{dm}{dt} \]
Where:
- \( T \) is the thrust force,
- \( v \) is the exhaust speed of the gas,
- \( \frac{dm}{dt} \) is the mass flow rate of the gas.

### Step 3: Calculate the total thrust required
To achieve an upward acceleration of \( a \), the thrust must overcome both the weight of the rocket and provide the necessary upward force. Thus, we can write:
\[ T = mg + ma \]
Where:
- \( m \) is the mass of the rocket (1000 kg),
- \( g \) is the acceleration due to gravity (approximately 10 m/s²),
- \( a \) is the upward acceleration (30 m/s²).

### Step 4: Substitute values into the thrust equation
Substituting the values into the thrust equation:
\[ T = (1000 \, \text{kg})(10 \, \text{m/s}^2) + (1000 \, \text{kg})(30 \, \text{m/s}^2) \]
\[ T = 10000 \, \text{N} + 30000 \, \text{N} \]
\[ T = 40000 \, \text{N} \]

### Step 5: Relate thrust to mass flow rate
Now, we can set the thrust equation equal to the thrust calculated:
\[ 40000 \, \text{N} = 500 \, \text{m/s} \cdot \frac{dm}{dt} \]

### Step 6: Solve for the mass flow rate (dm/dt)
Rearranging the equation to solve for \( \frac{dm}{dt} \):
\[ \frac{dm}{dt} = \frac{40000 \, \text{N}}{500 \, \text{m/s}} \]
\[ \frac{dm}{dt} = 80 \, \text{kg/s} \]

### Conclusion
The amount of gas ejected per second to supply the needed thrust is \( 80 \, \text{kg/s} \).

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