A space probe is traveling in outer space with a momentum that has a magnitude of `7.5 xx 10^7` kg-m/s. A retrorocket is fired to slow down the probe. It applies a force to the probe that has a magnitude of `2.0 xx 10^6` N and a direction opposite to the probe's motion. It fires for a period of 12 s. Determine the momentum of the probe after the retrorocket ceases to fire.

Options

A. `+5.1xx10^7` kg.m/s
B. `+2.4xx10^7` kg.m/s
C. `+7.5xx10^7` kg.m/s
D. `+9.9xx10^7` kg.m/s

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the initial momentum of the probe
The initial momentum \( p_i \) of the probe is given as:
\[ p_i = 7.5 \times 10^7 \, \text{kg m/s} \]

### Step 2: Identify the force applied by the retrorocket
The force \( F \) applied by the retrorocket is:
\[ F = 2.0 \times 10^6 \, \text{N} \]
Since the force is applied in the opposite direction to the motion of the probe, we can consider it as negative in our calculations.

### Step 3: Calculate the change in momentum
The change in momentum \( \Delta p \) can be calculated using the formula:
\[ \Delta p = F \cdot \Delta t \]
where \( \Delta t \) is the time duration for which the force is applied. Given that the force is applied for \( \Delta t = 12 \, \text{s} \):
\[ \Delta p = - (2.0 \times 10^6 \, \text{N}) \cdot (12 \, \text{s}) = - 2.4 \times 10^7 \, \text{kg m/s} \]
(Note: The negative sign indicates that the momentum is decreasing.)

### Step 4: Calculate the final momentum
The final momentum \( p_f \) of the probe can be calculated using the initial momentum and the change in momentum:
\[ p_f = p_i + \Delta p \]
Substituting the values:
\[ p_f = 7.5 \times 10^7 \, \text{kg m/s} - 2.4 \times 10^7 \, \text{kg m/s} \]
\[ p_f = (7.5 - 2.4) \times 10^7 \, \text{kg m/s} = 5.1 \times 10^7 \, \text{kg m/s} \]

### Step 5: Conclusion
The final momentum of the probe after the retrorocket ceases to fire is:
\[ \boxed{5.1 \times 10^7 \, \text{kg m/s}} \]