A 500 kg rocket is set for a vertical firing. If the exhaust speed is 1000 m/s, then the mass of gas ejected per second to supply the thrust needed to overcome the weight of the rocket is (Take g = `10 m//s^(2)`)

Options

A. 2 kg/s
B. 3 kg/s
C. 5 kg/s
D. 7 kg/s

Answer

To solve the problem, we need to determine the mass of gas ejected per second by the rocket to supply the thrust needed to overcome its weight. Here’s a step-by-step solution:

### Step 1: Identify the weight of the rocket
The weight \( W \) of the rocket can be calculated using the formula:
\[ W = m \cdot g \]
where:
- \( m = 500 \, \text{kg} \) (mass of the rocket)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating the weight:
\[ W = 500 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 5000 \, \text{N} \]

### Step 2: Understand the thrust force
The thrust force \( T \) generated by the rocket must be equal to the weight of the rocket in order for it to lift off. Therefore, we have:
\[ T = W = 5000 \, \text{N} \]

### Step 3: Use the thrust formula
The thrust force can also be expressed in terms of the mass flow rate of the exhaust gases and the exhaust speed:
\[ T = \frac{dm}{dt} \cdot v \]
where:
- \( \frac{dm}{dt} \) is the mass of gas ejected per second (which we need to find)
- \( v = 1000 \, \text{m/s} \) (exhaust speed)

### Step 4: Set the thrust equal to the weight
Since we know that the thrust must equal the weight, we can set up the equation:
\[ 5000 \, \text{N} = \frac{dm}{dt} \cdot 1000 \, \text{m/s} \]

### Step 5: Solve for \( \frac{dm}{dt} \)
Rearranging the equation to solve for \( \frac{dm}{dt} \):
\[ \frac{dm}{dt} = \frac{5000 \, \text{N}}{1000 \, \text{m/s}} = 5 \, \text{kg/s} \]

### Final Answer
The mass of gas ejected per second to supply the thrust needed to overcome the weight of the rocket is:
\[ \frac{dm}{dt} = 5 \, \text{kg/s} \]

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