A `3000 kg` space probe is moving in a gravity free space at a constant velocity of `300 m//s`. To change the direction of space probe, rockets have been fired in a direction perpendicular to the direction of initial motion of the space probe, the rocket firing exerts a thrust of `4000 N` for `225 s`. The space probe will turn by an angle of (neglect the mass of the rockets fired)

Options

A. `30^(@)`
B. `60^(@)`
C. `45^(@)`
D. `37^(@)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the initial momentum of the space probe
The initial momentum (P_initial) of the space probe can be calculated using the formula:
\[ P_{\text{initial}} = m \cdot v_i \]
where:
- \( m = 3000 \, \text{kg} \) (mass of the space probe)
- \( v_i = 300 \, \text{m/s} \) (initial velocity)

Calculating:
\[ P_{\text{initial}} = 3000 \, \text{kg} \cdot 300 \, \text{m/s} = 900000 \, \text{kg m/s} \]

### Step 2: Calculate the change in momentum due to the thrust
The change in momentum (P_thrust) caused by the thrust of the rockets can be calculated using:
\[ P_{\text{thrust}} = F \cdot t \]
where:
- \( F = 4000 \, \text{N} \) (thrust)
- \( t = 225 \, \text{s} \) (time duration)

Calculating:
\[ P_{\text{thrust}} = 4000 \, \text{N} \cdot 225 \, \text{s} = 900000 \, \text{N s} \]

### Step 3: Apply conservation of momentum in the y-direction
When the rockets are fired, the momentum in the y-direction (P_y) will equal the change in momentum due to the thrust:
\[ P_y = P_{\text{thrust}} = 900000 \, \text{kg m/s} \]

### Step 4: Apply conservation of momentum in the x-direction
The momentum in the x-direction (P_x) remains unchanged since there are no external forces acting in that direction:
\[ P_x = P_{\text{initial}} = 900000 \, \text{kg m/s} \]

### Step 5: Relate the momentum components to the angle θ
The momentum components can be expressed in terms of the angle θ:
- In the y-direction: \( P_y = P \sin(\theta) \)
- In the x-direction: \( P_x = P \cos(\theta) \)

Where \( P \) is the resultant momentum after the thrust is applied.

### Step 6: Set up the equations
From the conservation of momentum:
1. \( P \sin(\theta) = 900000 \)
2. \( P \cos(\theta) = 900000 \)

### Step 7: Divide the equations to find θ
Dividing the first equation by the second:
\[ \frac{P \sin(\theta)}{P \cos(\theta)} = \frac{900000}{900000} \]
This simplifies to:
\[ \tan(\theta) = 1 \]

### Step 8: Solve for θ
Taking the arctan of both sides:
\[ \theta = \arctan(1) = 45^\circ \]

### Final Answer
The space probe will turn by an angle of \( 45^\circ \).

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