A rocket is going upward with acceleration motion. A man sitting in it feels his weight increased 5 times 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 = 10 ms^(-2))`.

Options

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

Explanation

To solve the problem, we need to analyze the forces acting on the rocket and the man inside it.

### Step 1: Understand the Forces Acting on the Rocket
When the rocket is accelerating upwards, the man inside feels an increased weight due to the acceleration of the rocket. The apparent weight \( W' \) that the man feels is given by:

\[ W' = m(g + A) \]

where:
- \( m \) is the mass of the man,
- \( g \) is the acceleration due to gravity (10 m/s²),
- \( A \) is the upward acceleration of the rocket.

### Step 2: Relate the Increased Weight to the Original Weight
According to the problem, the man feels that his weight has increased 5 times his original weight. Therefore, we can write:

\[ W' = 5mg \]

### Step 3: Set Up the Equation
From the two equations we have:

\[ m(g + A) = 5mg \]

### Step 4: Simplify the Equation
We can simplify the equation by dividing both sides by \( m \) (assuming \( m \neq 0 \)):

\[ g + A = 5g \]

### Step 5: Solve for the Acceleration \( A \)
Now, we can isolate \( A \):

\[ A = 5g - g = 4g \]

Substituting \( g = 10 \, \text{m/s}^2 \):

\[ A = 4 \times 10 = 40 \, \text{m/s}^2 \]

### Step 6: Calculate the Total Force Exerted by the Rocket Engine
The total force \( F \) exerted by the rocket engine must overcome both the gravitational force and provide the upward acceleration. The total force can be calculated using Newton's second law:

\[ F = (m + M)g + (m + M)A \]

Where:
- \( M \) is the mass of the rocket (including the man), given as \( 1.0 \times 10^4 \, \text{kg} \).

### Step 7: Substitute Values into the Force Equation
We can substitute \( g \) and \( A \) into the equation:

\[ F = (1.0 \times 10^4)(10) + (1.0 \times 10^4)(40) \]

Calculating each part:

1. Gravitational force: \( (1.0 \times 10^4)(10) = 1.0 \times 10^5 \, \text{N} \)
2. Force due to acceleration: \( (1.0 \times 10^4)(40) = 4.0 \times 10^5 \, \text{N} \)

### Step 8: Calculate Total Force
Now, add the two forces together:

\[ F = 1.0 \times 10^5 + 4.0 \times 10^5 = 5.0 \times 10^5 \, \text{N} \]

### Final Answer
The force being applied by the rocket engine is \( 5.0 \times 10^5 \, \text{N} \).

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