A rocket is going upward with acceleration motion. A man string in it feels his weight increased 5 time his own weight. If the mass of the rocket including that of the man is `1.0xx10^(4)kg` , how much force is being applied by rocket engine? `(Take g=10ms^(-2))` .

Options

A. `1. 5xx10^(4)N`
B. 2. `5xx10^(5)N`
C. 3. `5xx10^(8)N`
D. 4. `2xx10^(4)N`

Explanation

To solve the problem, we need to find the force being applied by the rocket engine. Let's break it down step by step.

### Step 1: Understand the Problem
The rocket is accelerating upwards, and the man inside feels his weight increased to 5 times his own weight. We need to calculate the force exerted by the rocket engine.

### Step 2: Define Variables
- Let the mass of the rocket including the man be \( m = 1.0 \times 10^4 \, \text{kg} \).
- The acceleration due to gravity is \( g = 10 \, \text{m/s}^2 \).
- The increased weight felt by the man is \( 5 \times \text{weight of the man} \).

### Step 3: Calculate the Weight of the Man
The weight of the man can be expressed as:
\[ W = mg \]
Since the total mass of the rocket including the man is given, we can express the total weight as:
\[ W_{\text{total}} = mg = (1.0 \times 10^4 \, \text{kg}) \times (10 \, \text{m/s}^2) = 1.0 \times 10^5 \, \text{N} \]

### Step 4: Set Up the Equation
When the rocket accelerates upwards, the man feels an effective weight that is 5 times his actual weight. The effective weight can be expressed as:
\[ W_{\text{effective}} = 5mg \]
This means:
\[ W_{\text{effective}} = 5 \times (mg) \]

### Step 5: Relate Forces
The net force acting on the man when the rocket accelerates upwards can be expressed as:
\[ F_{\text{net}} = ma + mg \]
Where \( a \) is the acceleration of the rocket. Since the effective weight is 5 times the actual weight:
\[ ma + mg = 5mg \]

### Step 6: Solve for Acceleration
Rearranging the equation gives:
\[ ma = 5mg - mg \]
\[ ma = 4mg \]
Dividing both sides by \( m \):
\[ a = 4g \]
Substituting \( g = 10 \, \text{m/s}^2 \):
\[ a = 4 \times 10 = 40 \, \text{m/s}^2 \]

### Step 7: Calculate the Force Exerted by the Rocket Engine
The total force exerted by the rocket engine can be calculated using Newton's second law:
\[ F = m(a + g) \]
Substituting the values:
\[ F = (1.0 \times 10^4 \, \text{kg}) \times (40 \, \text{m/s}^2 + 10 \, \text{m/s}^2) \]
\[ F = (1.0 \times 10^4 \, \text{kg}) \times (50 \, \text{m/s}^2) \]
\[ F = 5.0 \times 10^5 \, \text{N} \]

### Final Answer
The force being applied by the rocket engine is:
\[ F = 5.0 \times 10^5 \, \text{N} \]

---