Uncategorized: Practice Problem & Solution

Solution Explained:

To solve this problem, we apply the core principles of Uncategorized. Understanding the underlying formula is key to arriving at the correct answer below:

The force on the satellite due to dust accretion is \( F = v_{text{rel}} \frac{dM}{dt} \). Here, the dust is stationary, so \( v_{text{rel}} = -v \). So, \( F = (-v) \frac{dM}{dt} \). From Newton's second law, \( F = M \frac{dv}{dt} \). Thus, \( M \frac{dv}{dt} = -v \frac{dM}{dt} \). Substituting \( \frac{dM}{dt} = \alpha v \), we get \( M \frac{dv}{dt} = -v (\alpha v) = -\alpha v^2 \). Therefore, \( \frac{dv}{dt} = -\frac{\alpha v^2}{M} \).