A satellite in a force - free space sweeps stationary interplanetary dust at a rate `dM//dt = alpha v`, where `M` is the mass , `v` is the velocity of the satellite and `alpha` is a constant. What is the deacceleration of the satellite ?

Options

A. `-(2alphav^(2))/(M)`
B. `-(alphav^(2))/(M)`
C. `-(alphav^(2))/(2M)`
D. `-alphav^(2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We know that the satellite is sweeping interplanetary dust at a rate given by:
\[ \frac{dM}{dt} = \alpha v \]
where \( M \) is the mass of the satellite, \( v \) is its velocity, and \( \alpha \) is a constant.

### Step 2: Apply Newton's Second Law
According to Newton's second law, the force acting on the satellite can be expressed as:
\[ F = m \frac{dv}{dt} + v \frac{dm}{dt} \]
Here, \( m \) is the mass of the satellite, \( \frac{dv}{dt} \) is the acceleration, and \( \frac{dm}{dt} \) is the rate of change of mass.

### Step 3: Substitute the Rate of Change of Mass
We substitute \( \frac{dM}{dt} = \alpha v \) into the equation:
\[ F = m \frac{dv}{dt} + v (\alpha v) \]
This simplifies to:
\[ F = m \frac{dv}{dt} + \alpha v^2 \]

### Step 4: Set the Force to Zero
Since the satellite is in a force-free space, we can set the force \( F \) to zero:
\[ 0 = m \frac{dv}{dt} + \alpha v^2 \]

### Step 5: Rearranging the Equation
Rearranging the equation gives:
\[ m \frac{dv}{dt} = -\alpha v^2 \]

### Step 6: Solve for Acceleration
Now, we can express acceleration \( a \) (where \( a = \frac{dv}{dt} \)):
\[ \frac{dv}{dt} = -\frac{\alpha v^2}{m} \]
This indicates that the satellite is experiencing a deceleration.

### Final Answer
Thus, the deceleration \( a \) of the satellite is given by:
\[ a = -\frac{\alpha v^2}{m} \]
The negative sign indicates that it is a deceleration.

---